Homelessness and Behavioral Health Care Coordination Act of 2023 This bill requires the Department of Housing and Urban Development (HUD) to award competitive grants to improve coordination of health care and homelessness services for individuals who are homeless, have significant behavioral health issues (e.g., mental health and substance use disorders), and voluntarily seek assistance. Entities eligible for these five-year grants include local and tribal governments, public housing agencies that administer housing choice vouchers, and certain nonprofits. The bill prohibits grant funds from being used to pay for rent or health care, except to increase the availability of Naloxone (i.e. medication that reverses opioid overdoses) and provide training for its administration. Further, the bill requires HUD to establish an interagency working group to (1) consult on the grant program, and (2) develop and circulate materials for providers of health care and homelessness services that simplify the services each offers in order to help each sector better understand the other.