We have recently hired a full-time caregiver for mom. Do I need to pay Social Security for her?

I agree with JessieBelle. One more step may be to check with your state employment agency to see what the laws are. If she is your employee, as opposed to an independent contractor, you may have to pay into workers compensation, as well. It's good to know all of this ahead of time to avoid potential problems.
Carol

Did you contract her as a self-employed person or as a personal employee? If she is self employed, then she will be responsible for paying her own SS and taxes quarterly. Otherwise, you will be responsible. This is something that you need to work with your caregiver to get clarification.