Can I get goverment assitance to help payfor a board & care home for my mom that has Alzheimer's? Her only income is Social security

Try your state website. Type "aging" into the search box. Look for links that can take you to funding. You may want to find your LTC ombusdsman on the site or go to www.ltcombudsman.org and type in the zip code. They are very knowledgeable about options. So much depends on where you live (or rather your mother lives).
Good luck,
Carol