This phase II trial studies how well nivolumab works, with or without BMS986205, in treating patients with stage II-IV squamous cell cancer of the head and neck. Immunotherapy with monoclonal antibodies, such as nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. BMS986205 may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Giving nivolumab with BMS986205 may work better than nivolumab alone in treating patients with squamous cell cancer of the head and neck.

PRIMARY OBJECTIVES: I. To assess the impact of IDO1 inhibitor BMS-986205 (BMS986205) and nivolumab versus nivolumab alone on tumor radiographic response both at the primary tumor site and in regional lymph nodes by investigator assessment at 5 weeks. SECONDARY OBJECTIVES: I. To investigate whether adding the IDO1 - inhibitor, BMS986205, to nivolumab therapy affects intratumoral and systemic anti-tumor immunity. II. To assess the impact of BMS986205 and nivolumab verses nivolumab alone on pathologic treatment effect bother at the primary and regional lymph nodes. III. To determine the effect of BMS986205 and nivolumab versus nivolumab alone on immune cell composition within the tumor microenvironment including the presence of effector T cells (Teff), regulatory T cells (Treg), and tumor-associated macrophages (TAM). IV. To further characterize the effect of BMS986205 when combined with nivolumab on kynurenine production and correlate these levels with effects on immune cell composition and polarization. V. To review the relationship of p16 status by immunohistochemistry with immune cell polarization, tumor radiographic response, and immune cell composition. VI. To review the relationship of PD-L1 status by immunohistochemistry with immune cell polarization, tumor radiographic response, and immune cell composition. VII. To assess the safety and tolerability of BMS986205 and nivolumab. VIII. Evaluate surgical wound healing post treatment. EXPLORATORY OBJECTIVES: I. To further characterize the effect of BMS986205 and nivolumab versus nivolumab alone through analysis of T cell repertoire. II. To assess the interactions between the immune and metabolic microenvironment through analysis of alterations in exosome composition in peripheral blood as it related to immune, cytokine and metabolic alterations before, during and after treatment. III. To identify risks for poor physical and mental health outcomes; examine bio-behavioral factors associated with cancer treatment outcomes; and evaluate the physical and psychosocial needs of cancer survivors through patient reported outcomes. OUTLINE: Patients are randomized to 1 of 2 arms. ARM I: Patients receive IDO1 inhibitor BMS-986205 orally (PO) once daily (QD). Beginning week 2, patients also receive nivolumab intravenously (IV) over 30 minutes on day 1. Treatment repeats for up to 5 weeks in the absence of disease progression or unacceptable toxicity. Patients showing a treatment response receive IDO1 inhibitor BMS-986205 PO QD for 4 additional weeks and receive nivolumab IV over 30 minutes on day 1, then undergo surgery at week 10. Those without a treatment response after 5 weeks undergo surgery within 7 days. ARM II: Patients receive nivolumab IV over 30 minutes on day 1 in the absence of disease progression or unacceptable toxicity. Patients showing treatment response after 4 weeks receive nivolumab IV over 30 minutes on day 1, then undergo surgery at week 9. Those without a treatment response after 4 weeks undergo surgery within 7 days. After completion of study treatment, patients are followed up periodically for 12 months.