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?

Hello,
The term for this is hair allograft and it cannot be performed because hair contains individualized cellular markers that will be rejected by the immune system system of the new host. In order to achieve this, your mom would have to go under immunosuppressant drugs - the side effects would greatly outweigh any advantage that a hair transplant may bring. Research this site to find alternative solutions to solve your mom’s hair-loss.

Other than between identical twins, hair follicles cannot be safely transplanted from one person to another.