National Institute for Research on Safety and Quality Act This bill establishes the National Institute for Research on Safety and Quality within the National Institutes of Health and transfers the authorities, duties, obligations, and related legal and administrative functions of the Agency for Healthcare Research and Quality to that institute. In addition, the new institute must promote health care quality improvement and conduct research and demonstration projects focused on (1) the delivery of health care in inner cities and rural and frontier areas, and (2) health care for priority populations such as low-income groups and the elderly.