Department of Veterans Affairs Quality Employment Act of 2017 This bill provides that the annual determination of the Veterans Health Administration (VHA) staff shortages shall include shortages for five clinical and five non-clinical occupations for each network. (Such determination currently includes the five occupations for which there are the largest Department of Veterans Affairs [VA]-wide shortages.) The bill establishes in the VA an executive management fellowship program to provide eligible employees of: (1) the Veterans Benefits Administration (VBA) and the VHA with private sector training and experience, and (2) a private-sector entity with VA training and experience. The VA shall: conduct an annual performance plan for each political appointee that is similar to the plan conducted for VA career senior executive employees; establish a single database that lists each vacant VA position that is critical, difficult to fill, or both, and each vacant mental health position; prescribe regulations to allow for excepted service appointments of qualifying students and recent graduates; provide VHA human resources with training on employee recruitment and retention; establish a promotional track system for employees who are technical experts to advance without being required to transition to management positions; develop a program to transition military medical people into VHA employment; develop a plan to hire qualified directors for each medical center without a permanent director; develop a standardized VA exit survey; and ensure that a recruiter for each Veterans Integrated Service Network visits, at least annually, each allopathic and osteopathic teaching institution with a graduate medical education program within the network to recruit individuals to be appointed to VHA positions. The VA may non-competitively appoint a qualified former career or career conditional employee to any VA position within the competitive service that is one grade higher than the position most recently occupied by the employee.