There are many ways of quilting in the hoop. You can also embroider on your fabric and batting, this way you may not need stabilizer. The backing is added later.
1. You can embroider squares and quilt each block 'sandwich'. The backing square is usually ½" larger (trimmed after all sewing is done) to allow for joining. When you join a row of blocks, you can cover the join with a strip perhaps 2"wide, (vintage quilts look fabulous with lace strips).
2. Join a row of embroidered blocks (with or without batting) then cut the backing the same size, tack on at the joins, and quilt. You then lay the next row on top, face down, backing on the bottom, and sew together. You would then add in the batting, tack down and quilt.
The quilt needs to be supported so it doesn't drag on the embroidery unit, which is why I prefer the first option. I won't even bother trying to embroider the middle sections of a quilt because of all the bulk in the throat area.
The queen of QAYGo I think, is Jenny Haskins.com, who has a lot of quilts and cd's with her son Simon. In USA, Margaret Moorehead is an agent for them. There are a few others.
Hope this helps you, I have seen some beautiful embroidered quilts done this way.