Nivolumab and BMS986205 in Treating Patients With Stage II-IV Squamous Cell Cancer of the Head and Neck

Inclusion Criteria: * Pathologically confirmed head and neck squamous cell carcinoma (HNSC). * Any stage 2 or greater HNSCC (American Joint Committee on Cancer \[AJCC\] 8th edition) of the 1) oral cavity, 2) larynx, 3) hypopharynx, 4) nasal cavity/paranasal sinuses or 5) stage 1 oropharyngeal with lymphadenopathy. Patients with resectable disease that is amenable to surgery are eligible. Patient must have been determined to be candidates for surgical resection by a multi-disciplinary team including a surgeon, a medical oncologist * Eastern Cooperative Oncology Group (ECOG) performance status 0-2. * White blood cells 2000/ul or more. * Absolute neutrophil count 1500/ul or more. * Platelets 100,000/ul or more. * Hemoglobin 9 g/dl or more. * Bilirubin less than or equal to 1.5 x the upper limit of normal (except subjects with Gilbert syndrome, who can have total bilirubin \< 3 mg/dl). * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) less than or equal to 3 x the upper limit of normal. * Glomerular filtration rate (GFR) greater than or equal to 40 ml/min using the Cockcroft-Gault formula or serum creatinine less than or equal to 1.5 x ULN. * Women of child bearing potential (WOCBP) should have a negative serum or urine pregnancy test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 21 days of study enrollment. * WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study treatment(s) plus 5 months post-treatment completion. * Males who are sexually active with WOCBP must agree to follow instructions for method(s) of contraception for the duration of treatment with study treatment(s) plus 7 months post-treatment completion. In addition, male participants must be willing to refrain from sperm donation during this time. * Males who are sexually active with WOCBP must agree to use a condom during any sexual activity for the duration of treatment with study treatment plus 7 months after the last dose of the study treatment (i.e., 90 days \[duration of sperm turnover\] plus the time required for nivolumab to undergo approximately 5 half-lives). This criterion applies to azoospermic males as well. In addition, male participants must be willing to refrain from sperm donation during this time. * Male mandatory condom use is regardless of whether the participant has undergone a successful vasectomy or if the female partner is pregnant. * Investigators shall counsel WOCBP, and male participants who are sexually active with WOCBP, on the importance of pregnancy prevention and the implications of an unexpected pregnancy. Investigators shall advise on the use of highly effective methods of contraception, which have a failure rate of \< 1% when used consistently and correctly. Hormonal contraceptives are prohibited methods of contraception for participants receiving BMS-986205 this study who are WOCBP. * All subjects must be able to comprehend and sign a written informed consent document. Exclusion Criteria: * Patients with nasopharyngeal carcinoma, salivary gland or skin primaries. * Patients with recurrent head and neck cancer treated previously with chemotherapy, radiation or immunotherapy. * Any history of a severe hypersensitivity reaction to any monoclonal antibody. * Any history of allergy to the study drug components. * Participants with a personal or family (i.e., in a first-degree relative) history or presence of cytochrome b5 reductase deficiency (previously called methemoglobin reductase deficiency) or other diseases that puts them at risk of methemoglobinemia. All participants will be screened for methemoglobin levels prior to randomization. * Participants with a history of G6PD deficiency or other congenital or autoimmune hemolytic disorders. All participants will be screened for G6PD deficiency prior to randomization. * Participants with history of serotonin syndrome. * Participants with active interstitial lung disease (ILD)/pneumonitis or history of ILD/ pneumonitis requiring steroids. * Prior treatment with BMS-986205 or any other IDO1 inhibitors. * Quantitative or qualitative G6PD assay results suggesting underlying G6PD deficiency. * Blood methemoglobin \> upper limit of normal (ULN), assessed in an arterial or venous blood sample or by co-oximetry. * History or presence of hypersensitivity or idiosyncratic reaction to methylene blue. * History of allergy or hypersensitivity to any study treatment components, specifically to that of BMS-986205. * Any concurrent malignancies; exceptions include- basal cell carcinoma of the skin, squamous cell carcinoma of the skin, superficial bladder cancer or in situ cervical cancer that has undergone potentially curative therapy. Patients with a history of other prior malignancy must have been treated with curative intent and must have remained disease-free for 2 years post-diagnosis. * Any diagnosis of immunodeficiency or receiving systemic steroid therapy (\> 10 mg daily prednisone equivalents) or any other form of immunosuppressive therapy within 14 days of initiation of therapy. * Patients that have an active autoimmune disease requiring systemic treatment within the past 3 months or a documented history of clinically severe autoimmune disease, or a syndrome that requires systemic steroids (\> 10 mg daily prednisone equivalents) or immunosuppressive agents. Subjects with vitiligo, type I diabetes mellitus, or resolved childhood asthma/atopy would be an exception to this rule. Subjects that require intermittent use of bronchodilators or local steroid injections would not be excluded from the study. Subjects with hypothyroidism stable on hormone replacement or Sjogren's syndrome will not be excluded from the study. * Patients must not be receiving any other investigational agents. * Patients with uncontrolled intercurrent illnesses including, but not limited to an active infection requiring systemic therapy or a known psychiatric or substance abuse disorder(s) that would interfere with cooperation with the requirements of the trial. * Patients must not be pregnant or breastfeeding. * Patients with a known human immunodeficiency virus infection (HIV 1/2 antibodies) or acquired immunodeficiency syndrome (HIV/AIDS), active hepatitis B (e.g., hepatitis B surface antigen \[HBsAg\] reactive) or hepatitis C (e.g., hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] \[qualitative\] is detected). * Patients with any evidence of current interstitial lung disease (ILD) or pneumonitis. * Patients with prior history of ILD or non-infectious pneumonitis that required steroids. * Patients who have received a live vaccine within 30 days of the planned start of study therapy.