I would verify that the spots are not rust (and that the fibers are not damaged). If it is "just dirt" I would air them out and finish the quilt. Then I would deal with washing and spot treating.
My reason for saying this: I have an old quilt top from my MIL and it was aweful. So I washed it - carefully in the tub. Yes, it is clean BUT the seams are wads of threads and I will need to do some serious trimming and pressing before I can work on it.