By fleece you mean the polyester microfiber fabric, yes? Not flannel?
You can use the fleece as the backing for the quilt. Lots of quilters do this.
The cotton fabric can be incorporated into a pieced top. To feature the cotton fabric, you might want to use a Warm Wishes pattern; you would just need to add three fabrics (could be solids) to act as frames around the larger cotton pieces. If you Google "warm wishes quilt" and then click on "images", you will see lots of examples. This is an easy quilt to make too.