I'm awake and brave at 3:25 so...
I'd start looking (Google) for textile museums in your county (or your grandmother's county) and then state, then part of the country.
I'd check with local historical societies - town or city, county state and see what they collect and what happens to their collections after a few years.
I might offer to frame the quilt under glass as a gift to a local hospital as a decoration. Offer it to a library with wall space and maybe would still frame it under glass.
Talk with other museums in your area, farm machinery museum or pioneer museum might love having a handmade quilt on it's walls.
Talking with these kinds of people might get you pointed in the right direction as well. Maybe they would know exactly who would be interested and so on.
I'd be sure to have the biographical information sewn onto the back somehow. IMHO the quilts would lose value if they lost their identity.
Good luck!
Jois