Yes, the grandparent is exempt from FICA tax, if she babysits in your home. She still needs to pay income tax, since you are using the FSA account (or if you were claiming the dependent care credit).
You do not need to do 'nanny tax" paperwork (the grandma exception).
For details on how your mother reports her income, see https://ttlc.intuit.com/community/tax-credits-deductions/discussion/i-m-retired-and-on-ss-i-took-car...
