Maternal Health Quality Improvement Act of 2021 This bill requires the Department of Health and Human Services (HHS) to undertake activities to improve maternal health, including in rural areas. Specifically, HHS must award grants for (1) developing best practices to reduce preventable maternal mortality, and (2) training health care professionals to reduce and prevent discrimination when providing maternal health services. Additionally, HHS may award grants for integrated maternal and infant health care services. The bill also provides statutory authority for grants that support perinatal quality collaboratives, which are networks of health care and public health professionals who work to improve pregnancy outcomes. Furthermore, the bill requires a study on efforts in health professional schools and training programs to reduce and prevent discrimination in the provision of maternal health services. To address care in rural areas, the bill (1) provides statutory authority for grants that support networks of rural providers to improve and coordinate maternal care, (2) makes maternal care providers eligible for certain telehealth grants, and (3) establishes a demonstration project to improve training on maternal care in rural areas through health professional schools and training programs. The bill also expands the maternal care responsibilities of specified HHS offices and programs.