Inclusion Criteria:

          1. Is a male or female participant aged 18 years or older.

          2. Has eastern Cooperative Oncology Group (ECOG) performance status 0 or 1.

          3. Female participants who:

               -  Are postmenopausal for at least 1 year before the Screening visit, or

               -  Are surgically sterile, or

               -  If childbearing potential, agree to practice 2 effective methods of
                  contraception, at the same time, from the time of signing the informed consent
                  through 180 days after the last dose of study drug, or

               -  Agree to practice true abstinence, when this is in line with the preferred and
                  usual lifestyle of the participants. (Periodic abstinence [example, calendar,
                  ovulation, symptothermal, postovulation methods] and withdrawal are not
                  acceptable methods of contraception.)

             Male participants, even if surgically sterilized (that is, status postvasectomy), who:

               -  Agree to practice effective barrier contraception during the entire study
                  treatment period and through 180 days after the last dose of study drug, or

               -  Agree to practice true abstinence, when this is in line with the preferred and
                  usual lifestyle of the participants. (Periodic abstinence [example, calendar,
                  ovulation, symptothermal, postovulation methods for the female partner] and
                  withdrawal are not acceptable methods of contraception.)

          4. Voluntary written consent must be given before performance of any study-related
             procedure not part of standard medical care, with the understanding that consent may
             be withdrawn by the participants at any time without prejudice to future medical care.

          5. Suitable venous access for the study-required blood sampling, including PK and
             pharmacodynamic (PD) sampling.

          6. Clinical laboratory values and other measures as specified below within 28 days before
             the first dose of study drug:

               -  Total bilirubin must be <=1.5*the upper limit of normal (ULN).

               -  Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) must be
                  <=2.5*ULN.

               -  Creatinine clearance must be greater than or equal to (>=) 60 milliliter per
                  (mL/) minute as estimated by the Cockcroft Gault equation or based on urine
                  collection (12 or 24 hours).

               -  Hemoglobin must be >=8 gram per deciliter (g/dL), absolute neutrophil count (ANC)
                  must be >=1500 per microliter (/mcL), and platelet count must be >=75,000/mcL.

               -  Lipase must be <=1.5*ULN and amylase <=1.5*ULN with no clinical symptoms
                  suggestive of pancreatitis and cholecystitis.

               -  Blood pressure <=Grade 1 (hypertensive participants are permitted if their blood
                  pressure is controlled to <=Grade 1 by hypotensive medications and glycosylated
                  HbA1C <=6.5%).

          7. Recovered (that is, <=Grade 1 toxicity) from the reversible effects of prior
             anticancer therapy.

          8. To be enrolled in the dose escalation phase of the study, participants must have a
             radiographically or clinically evaluable tumor, but measurable disease as defined by
             RECIST version 1.1 is not required for participation in this study.

          9. To be enrolled in the TNBC expansion cohort, participants must have:

               -  Histologically confirmed, metastatic TNBC with measurable disease per RECIST
                  version 1.1.

               -  Triple-negative disease (estrogen receptor, progesterone receptor, and human
                  epidermal growth factor receptor 2 (HER2) negativity confirmed on a histological
                  biopsy of a metastatic tumor lesion (receptor conversion not allowed).

               -  Safely accessible tumor lesions (based on investigator's assessment) for serial
                  pre treatment and post treatment biopsies are required for participants receiving
                  TAK-659 monotherapy run-in treatment for 2 weeks followed by TAK-659 plus
                  nivolumab combination treatment [ approximately 10/30 response-evaluable
                  participants]; adequate, newly obtained, core or excisional biopsy of a
                  metastatic tumor lesion not previously irradiated is required. Mandatory biopsies
                  will be taken before TAK-659 monotherapy, after the 2 weeks of TAK-659
                  monotherapy, and after 6 weeks of TAK-659 plus nivolumab combination therapy. An
                  optional biopsy may be taken at PD with additional consent from the participants.

               -  One, 2, or 3 prior lines of chemotherapy for metastatic disease and with
                  progression of disease on last treatment regimen.

               -  For the purposes of this study, neoadjuvant and/or adjuvant chemotherapy regimens
                  do not count as a prior line of therapy.

               -  Prior treatment must include an anthracycline and/or a taxane in the neoadjuvant,
                  adjuvant, or metastatic setting with the exception for participants who are
                  clinically contraindicated for these chemotherapies.

         10. To be enrolled in the NSCLC expansion cohort, participants must have:

               -  Locally advanced or metastatic (stage IIIB, stage IV, or recurrent) NSCLC with
                  measurable lesions per RECIST version 1.1.

               -  PD during or following at least 1 prior treatment. Participants should have
                  received a prior platinum-based 2-drug regimen for locally advanced,
                  unresectable/ inoperable or metastatic NSCLC or disease recurrence within 6
                  months of treatment with a platinum-based adjuvant/neoadjuvant regimen or
                  combined modality (example, chemoradiation) regimen with curative intent.

               -  Participants with epidermal growth factor receptor (EGFR) or anaplastic lymphoma
                  kinase (ALK) genomic alternations should have PD on prior United States (US) Food
                  and Drug Administration (FDA) approved therapy for these aberrations.

               -  Safely accessible tumor lesions (based on investigator's assessment) for serial
                  pretreatment and posttreatment biopsies are required for participants receiving
                  TAK-659 monotherapy run-in treatment for 2 weeks followed by TAK-659 plus
                  nivolumab combination treatment (approximately10/30 response-evaluable
                  participants); adequate, newly obtained, core or excisional biopsy of a
                  metastatic tumor lesion not previously irradiated is required. Mandatory biopsies
                  will be taken before TAK-659 monotherapy, after the 2 weeks of TAK-659
                  monotherapy, and after 6 weeks of TAK-659 plus nivolumab combination therapy. An
                  optional biopsy may be taken at progression with additional consent from the
                  participants.

         11. To be enrolled in the HNSCC expansion cohort, participants must have:

               -  Histologically confirmed recurrent or metastatic HNSCC (oral cavity, pharynx, or
                  larynx) that is stage III/IV and not amenable to local therapy with curative
                  intent (surgery or radiation therapy with or without chemotherapy).

               -  Histologically confirmed recurrent or metastatic squamous cell carcinoma of
                  unknown primary or nonsquamous histologies (example, mucosal melanoma) are not
                  allowed.

               -  Histologically confirmed recurrent or metastatic carcinoma of the nasopharynx is
                  allowed, but these participants will not be included as response-evaluable
                  participants for efficacy analysis of HNSCC.

               -  Measurable disease per RECIST version 1.1.

               -  Tumor progression or recurrence within 6 months of the last dose of
                  platinum-based therapy in the adjuvant (that is, with radiation after surgery),
                  primary (that is, with radiation), recurrent, or metastatic setting.

               -  Safely accessible tumor lesions (based on investigator's assessment) for serial
                  pretreatment and posttreatment biopsies are required for participants receiving
                  TAK-659 monotherapy run-in treatment for 2 weeks followed by TAK-659 plus
                  nivolumab combination treatment (approximately 10/30 response-evaluable
                  participants); adequate, newly obtained, core or excisional biopsy of a
                  metastatic tumor lesion not previously irradiated is required. Mandatory biopsies
                  will be taken before TAK-659 monotherapy, after the 2 weeks of TAK-659
                  monotherapy, and TAK-659 after 6 weeks of TAK-659 plus nivolumab combination
                  therapy. An optional biopsy may be taken at progression with additional consent
                  from the participants.

        Exclusion Criteria:

          1. Has active brain metastases or leptomeningeal metastases.

          2. Has active, or suspected autoimmune disease or a history of known autoimmune disease,
             with the exception of:

             o Participants with vitiligo, type I diabetes mellitus, resolved childhood
             asthma/atopy, residual hypothyroidism due to autoimmune condition requiring only
             hormone replacement, psoriasis not requiring systemic treatment, or conditions not
             expected to recur in the absence of an external trigger.

          3. Any condition requiring systemic treatment with corticosteroids (less than [>]10 mg
             daily prednisone equivalents) or other immunosuppressive medications within 14 before
             first dose of study drug.

             o Corticosteroids for topical use or in nasal spray are allowed, as are inhaled
             steroids and adrenal replacement steroid doses >10 mg daily in the absence of active
             autoimmune disease.

          4. Has history of pneumonitis requiring treatment with steroids; history of idiopathic
             pulmonary fibrosis, drug-induced pneumonitis, organizing pneumonia, or evidence of
             active pneumonitis on the Screening chest computed tomography scan (CT scan); history
             of radiation pneumonitis in the radiation field (fibrosis) is permitted.

          5. Has history of interstitial lung disease.

          6. Prior therapy with experimental antitumor vaccines; any T-cell co-stimulation agents
             or inhibitors of checkpoint pathways, such as anti- programmed cell death protein 1
             (PD-1), anti- programmed cell death 1 ligand 1 (PD-L1), anti- programmed cell death 1
             ligand 2 (PD-L2), anti-CD137, or anti-CTLA-4 antibody; or other agents specifically
             targeting T cells are prohibited. However, for dose escalation, prior treatment with
             the marketed inhibitors of the immune checkpoint pathway, such as nivolumab and
             pembrolizumab, is allowed. In addition, in each of the expansion cohorts, 6
             response-evaluable participants with prior exposure to anti-PD-1 or anti-PD-L1 agents
             will be allowed to enroll.

          7. Has any serious medical or psychiatric illness, including drug or alcohol abuse, that
             could, in the investigator's opinion, potentially interfere with the completion of
             treatment according to this protocol.

          8. Has life-threatening illness unrelated to cancer.

          9. Is female participant who are lactating and breast-feeding or a positive serum
             pregnancy test during the Screening period or a positive urine pregnancy test on Day 1
             before the first dose of study drug.

         10. Systemic anticancer treatment including investigational agents or radiotherapy <2
             weeks before the first dose of study treatment (<4 weeks for antibody-based therapy
             including unconjugated antibody, antibody-drug conjugate, and bi-specific T-cell
             engager agents; <=8 weeks for cell-based therapy or antitumor vaccine) or have not
             recovered from acute toxic effects from prior chemotherapy and radiotherapy.

         11. Prior treatment with investigational agents =<21 days or =<5*their half-lives
             (whichever is shorter) before the first dose of study treatment. A minimum of 10 days
             should elapse from prior therapy to initiating protocol therapy.

         12. Major surgery within 14 days before the first dose of study drug and not recovered
             fully from any complications from surgery.

         13. Systemic infection requiring intravenous antibiotic therapy or other serious infection
             within 14 days before the first dose of study drug.

         14. Known human immunodeficiency virus (HIV) positive (testing not required).

         15. Known hepatitis B surface antigen-positive or known or suspected active hepatitis C
             infection (testing not required).

         16. Participants with another malignancy within 2 years of study start. Participants with
             nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have
             undergone complete resection and are considered disease-free at the time of study
             entry.

         17. Any clinically significant comorbidities, such as uncontrolled pulmonary disease,
             known impaired cardiac function or clinically significant cardiac disease (specified
             below), active central nervous system disease, active infection, or any other
             condition that could compromise the participant's participation in the study.

             Participants with any of the following cardiovascular conditions are excluded:

               -  Acute myocardial infarction within 6 months before starting study drug.

               -  Current or history of New York Heart Association Class III or IV heart failure

               -  Evidence of current uncontrolled cardiovascular conditions including cardiac
                  arrhythmias, angina, pulmonary hypertension, or electrocardiographic evidence of
                  acute ischemia or active conduction system abnormalities.

               -  Fridericia corrected QT interval (QTcF) >450 milliseconds (msec) (men) or >475
                  msec (women) on a 12-lead electrocardiogram (ECG) during the Screening period.

               -  Abnormalities on 12-lead ECG including, but not limited to, changes in rhythm and
                  intervals that in the opinion of the investigator are considered to be clinically
                  significant.

         18. Known gastrointestinal (GI) disease or GI procedure that could interfere with the oral
             absorption or tolerance of TAK-659 including difficulty swallowing tablets; diarrhea
             >Grade 1 despite supportive therapy.

         19. Use or consumption of any of the following substances:

               -  Medications or supplements that are known to be inhibitors of P-glycoprotein
                  (P-gp) and/or strong reversible inhibitors of cytochrome P450 (CYP) 3A within 5
                  times the inhibitor half-life (if a reasonable half-life estimate is known) or
                  within 7 days (if a reasonable half-life estimate is unknown) before the first
                  dose of study drug. In general, the use of these agents is not permitted during
                  the study.

               -  Non-oncology vaccine therapies for prevention of infectious diseases (example,
                  human papillomavirus [HPV] vaccine) within 4 weeks of study drug administration.
                  The inactivated seasonal influenza vaccine can be given to participants before
                  treatment and while on therapy without restriction. Influenza vaccines containing
                  live virus or other clinically indicated vaccinations for infectious diseases
                  (example, pneumovax, varicella) may be permitted but must be discussed with the
                  sponsor's medical monitor and may require a washout period before and after
                  administration of vaccine.

               -  Medications or supplements that are known to be strong CYP3A mechanism-based
                  inhibitors or strong CYP3A inducers and/or P-gp inducers within 7 days, or within
                  5 times the inhibitor or inducer half-life (whichever is longer), before the
                  first dose of study drug. The use of these agents is not permitted during the
                  study.

               -  Grapefruit-containing food or beverages within 5 days before the first dose of
                  study drug. Note that grapefruit-containing food and beverages are prohibited
                  during the study.

         20. For dose expansion participants who will have tumor biopsies collected:

               -  ECOG performance status >1.

               -  Activated partial thromboplastin time (aPTT) or plasma thromboplastin (PT)
                  outside the institution's standard of care.

               -  Platelet count <75,000/mcL.

               -  Known bleeding diathesis or history of abnormal bleeding, or any other known
                  coagulation abnormalities that would contraindicate the tumor biopsy procedure.

               -  Ongoing therapy with any anticoagulant or antiplatelet agents (example, aspirin,
                  clopidogrel, coumadin, heparin, or warfarin) that cannot be held to permit tumor
                  biopsy).