If the partners are not married, each will be treated as single when applying for Medicaid. It does not matter that they are in a relationship. But beware running afoul of the transfer rules. Married couples may freely transfer assets back and forth, but if an unmarried couple does so, it may be treated as a transfer of assets and penalized with a period of ineligibility for benefits based on the amount transferred.
Get more information on Medicaid asset transfer rules in this article.