Behavioral Health Infrastructure Improvement Act This bill amends title XIX (Medicaid) of the Social Security Act to temporarily increase the Federal Medical Assistance Percentage (FMAP) for behavioral health infrastructure and systems improvements under a state medical assistance program. For a period of five years, a state's FMAP for such improvements shall equal 90%. (Generally, a state's FMAP under Medicaid may range from 50% to 83%.) The bill also establishes a grant program for states to establish or support a behavioral health assistance office or ombudsman to coordinate behavioral health and other assistance for Medicaid beneficiaries. The office or ombudsman shall: (1) track, quantify, and report on problems and inquiries encountered by Medicaid beneficiaries with respect to access to behavioral health services; (2) educate Medicaid beneficiaries on their rights and responsibilities to such access; and (3) assist Medicaid beneficiaries in the enforcement of those rights and in accessing services.