Veterans Fellowship Act This bill requires the Veterans' Employment and Training Service within the Department of Labor to implement a pilot program under which states may use grants or contracts to carry out a short-term fellowship program. Each fellowship program must run for a period that does not exceed 20 weeks, provide a monthly stipend, and provide veterans an opportunity to be employed on a long-term basis following the fellowship. The Government Accountability Office must report on the pilot program within four years after it commences.