There is term called "knife edge" binding. It is a lot of work as it is usually done by hand. You square up the quilt and leave the last inch all around the edge unquilted. You then fold under the 1/4 inch seam allowance around the whole top. You then do the same to the back. When you have then perfect you hand stitch the two edges together with the invisible ladder stitch. You can trim the batt back 1/4 inch before you start the hand sewing but I usually just scrunch it in a little.
Ricky Tim's had a tutorial on using a facing on a quilt but that is typically hand sewn to the backing and it leaves a stitching line. If you match the fabric it is not very noticeable.