Advancing the Quality and Understanding of American Aquaculture Act or the AQUAA Act This bill establishes national standards for sustainable offshore aquaculture. It also establishes a national plan to identify and designate aquaculture opportunity areas. The Department of Commerce must develop a plan and time line to systematically (1) assess the exclusive economic zone, (2) prepare an inventory of sites suitable for aquaculture opportunity areas, and (3) designate aquaculture opportunity areas. Commerce must also 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. The bill directs Commerce to create an Office of 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.