Quality Care for Moms and Babies Act This bill directs the Centers for Medicare & Medicaid Services (CMS) to: identify and publish a recommended core set of maternal and infant quality measures for women and children, as specified by the bill; publish an initial core set of any such measures applicable to mothers and infants eligible under Medicaid or the Children's Health Insurance Program (CHIP); establish a Maternal and Infant Quality Measurement Program; and establish an online clearinghouse of resources for entities working to improve maternity and infant care quality. The CMS may make grants to eligible entities for: the development of new state and regional maternity and infant care quality collaboratives; expanded activities of existing collaboratives; and maternity and infant care initiatives within established state and regional quality collaboratives that are not focused exclusively on maternity care.