Yes, you can, and as already mentioned, be sure to tell the quilter what pieces are not backing.
The only place you may have a problem is at the bottom of the quilt. The bottom of the quilt will be determined by which way it is loaded on the frame. As a quilt is rolled onto the takeup bar (the bar that collects quilted parts of the quilt as it is completed), the backing rolls up a bit more than the quilt top. Think of rolling a magazine around it's spine and how the page edges don't line up---it's like that. You need more of the actual backing at the "bottom" than anywhere else for that reason. Many longarmmers, myself included, have learned this the hard way when they ran out of backing at the "bottom" before the quilt was done, and had to add more backing, which is not a fun job at that point. Sometimes this will influence which way I load a quilt, so I can have the most extra backing at the "bottom" where it is needed.
On the "sides" the extra backing is used for the tension clamps to clamp onto and not be in the way of quilting, so not as much extra backing is needed on the sides sometimes, depending on the particular side clamps the quilter uses.