Advancing the Quality and Understanding of American Aquaculture Act or the AQUAA Act This bill establishes national standards for sustainable offshore aquaculture. The bill directs the Department of Commerce to create an Office of Offshore Aquaculture within the National Oceanic and Atmospheric Administration to coordinate regulatory, scientific, outreach, and international issues related to aquaculture. The bill also establishes a unified permitting and review process for aquaculture operations. Commerce must establish a research and development grant program to promote research and extension activities, including to (1) create innovative design and engineering solutions to common obstacles within the aquaculture industry; (2) evaluate the role of genetics in relation to brood stock production, disease management, and interactions between cultured species and wild stocks; and (3) improve techniques for monitoring, assessing, and addressing environmental impacts of aquaculture and develop and evaluate methodologies to prevent, minimize, and mitigate potential adverse environmental impacts.