Where do I apply to get paid for taking care of my mom?

Check if your state has a COPES program, which can pay a family member, after a training program is completed. I don't think every state offers this, and there are probably some financial qualifiers, to be eligible if there is. It doesn't pay well, as most caregiver jobs don't, but it is Something! Good luck!

Your AREA AGENCY ON AGING (AAA), will probably have information on this.

Find the Area Agency on Aging office for your state and county. (Just Google Area Agency on Agency and your county). That office should be able to tell you if your state has any programs that pay caregivers. They are rare and often pay very little.
The other benefit of contacting your AAA is they will be able to connect you with what services are available in your area.

There really isn't anywhere to go to get paid as a caregiver. Those few of us that do get paid...the person paying them is their parents.   The vast majority are paid nothing.

If you are looking to set up a payment systems where your Mom pays you...get a caregiver agreement in writing first..signed by everyone. This is absolutely to protect both of you.

If you Mom cannot pay you, and she qualifies for Medicaid...there maybe a small payment (depending on your State) ... but those only pay a couple hours per week....nowhere near enough to pay your bills. You are far better off going and getting a job, if Medicaid will pay something....get someone to come in as respite for you.