Well you learned about washing before quilting the hard way! It can be done, but you have to fold it, pin it and put it in a mesh bag and then set your washer to agitate on gentle, then leave it pinned and folded when you put it in the dryer. That's actually more work than it's worth. I know because I did it once!
It sounds like you haven't actually put the batting or backing on it yet. If that's true, Why not use some scraps (if you have them), sew pieces of them (large enough to generously span the fray together), put fusible liner under the new pieces and applique them on over the frayed edges?