Maternal Health Quality Improvement Act This bill requires the Department of Health and Human Services (HHS) to establish specified grant programs to support positive maternal health care outcomes for pregnant and postpartum women and their infants. Specifically, HHS must award grants to (1) develop best practices to reduce preventable maternal mortality, and (2) train health care professionals to reduce and prevent discrimination when providing maternal health services. In addition, HHS may award grants to establish evidence-informed, integrated health care services to pregnant and postpartum women and their infants. The bill also provides statutory authority for a grant program that supports perinatal quality collaboratives. These are networks of perinatal care providers and public health professionals that work to improve pregnancy outcomes through continuous quality improvement. HHS also must (1) contract with an independent research organization to study and make recommendations on trainings for health professional schools and education programs on reducing and preventing discrimination in the provision of maternal health services, and (2) address pregnant and postpartum women and their infants in public awareness initiatives related to vaccinations.