I definitely would...and once given from the heart with the message that you are hoping that she will know that she can always turn to you everytime she lets the quilt warm her, it will be hers to keep or do with as she wishes.
She might have come from a home with a lack of lessons in keeping and appreciation, but she is apparently trying a new path to travel and has reached out to you in the journey, and deserves a chance at change.
If I gave a quilt (or anything I made) only with the assurance that it would be kept and cherished forever, it would mean that I was only thinking about how I felt in the gifting.
Make a simple one and fill it with love.