Thank you for the pictures. It is a lovely quilt.
If it had been mine - and it was a quilt that I was using a lot - I might have just either serged or zigzagged over the weak /fraying edge and called it "good enough" - just to keep it from fraying out. Not a pretty finish - but one I have used on the hems of jeans and bath towels.
But I think you are looking for something "nicer" - if the quilt is large enough - and the quilting stops far enough away from the edge - I would just fold it under an additional 1/4 inch (so instead of 1/4 inch you would have 1/2 inch) fold under - and sew the edges back together like it had been - assuming the fabric is strong enough.
Otherwise - if you can find a binding fabric suitable - I would fold those raw edges back in - baste the edges back together - and than use a 3 inch double fold binding - it would finish at about 1/2 inch - that would just encase the edges.