Would I be able to donate some of my hair follicles to my mom to get a hair transplant in case her own hair couldn't do the job?

You cannot donate your hair to your mom because the grafts will reject.  Only way to donor hair to another person is to your genetically identical twin.

I'm afraid not. Only identical twins can donate hair follicles to each other. Have your mother visit an expert at hair loss, and there may be other alternatives.
Thank you for your question, and best of luck to your mother.

Unfortunately the answer is no. Hair transplant is actually a tissue transplant, and would require anti-tissue rejection medication if donated from one person to a different person.