Witness Security and Protection Grant Program Act of 2013 - Directs the Attorney General to: (1) make competitive grants to state, tribal, and local governments to establish or maintain programs that provide protection or assistance to witnesses in court proceedings involving a homicide, a serious violent felony, a serious drug offense, gangs, or organized crime; (2) evaluate grant applicants based on specified criteria, including the prevalence of witness intimidation and the number of such crimes per capita in the applicant's jurisdiction; (3) provide technical assistance to applicants for establishing or maintaining such programs; and (4) develop and disseminate best practice models to assist such governments in addressing witness safety, short-term and permanent witness relocation, financial and housing assistance, and other necessary witness protection or assistance services. Urges such governments to use such models to evaluate, improve, and develop witness protection or witness assistance programs.