Quality Care for Moms and Babies Act This bill directs the Centers for Medicare & Medicaid Services (CMS) to take a series of actions relating to the quality of maternal and infant care under Medicaid, the Children's Health Insurance Program (CHIP), and other health programs. Specifically, the CMS must 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 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 also make grants to eligible entities (e.g., state Medicaid programs, health departments, and provider organizations) to develop or expand maternity and infant care quality collaboratives or initiatives.