Q. Will that require her to pay Social security and Medicare (FICA) taxes?
A. Simple answer: no. A grandparent who babysits in the parent's home is exempt from the "nanny tax:".
But taxes aren't always simple. Although she is exempt from FICA tax, that income is subject to income tax.
If Granny is your dependent, what you pay her is not eligible for FSA or the dependent care credit.
For more detail, see a similar question at https://ttlc.intuit.com/community/tax-credits-deductions/discussion/grandparent-babysitting-my-child...