Can my mother pay me to take care of her and at a later date apply for Medicaid?

There's a good primer on how to get paid for being a family caregiver at https://www.agingcare.com/articles/how-to-get-paid-for-being-a-caregiver-135476.htm

That site includes a link to another article about personal caregiving agreements, including a template for creating such.

I think what you are asking is will her paying you qualify as a use of funds to spend down, and qualify for Medicaid.  You will need a written agreement, charge fair rates, have the agreement signed by her and you, notarized, BEFORE you start charging.  If she is incompetent, her POA or guardian will need to sign.  If that is true, and you are the POA or guardian you may want to consult an attorney.  You will need to treat this as taxable income and include on your 1040.

Good insights provided below. More info would be helpful, like: how old is your mom? Does she have physical or cognitive issues right now? Maybe ask yourself if you can "afford" to be a paid caregiver? These are low wages with not much increase over time, no 401K, no paid healthcare, no paid vacation, etc. Also, who will care for your mother if you are sick? On vacation? Need a personal day off? Caregiving can be very isolating for both the giver and the receiver, especially when they become much less able to leave the home. Caregiver burnout is a real thing and on this forum there are thousands of posts regarding it. I'm not saying to not be a caregiver. Just suggesting go into it with your eyes very widely opened.