Most likely the backing fabric is stretching when you sew. I usually starch my backing fabric heavily to prevent this kind of problem. Since you have already sandwiched your quilt, you might want to try spray starching the backing several times. Starching stabilizes fabric so it doesn't stretch when you sew.
If your machine has a presser foot adjustment, you might also try reducing the amount of pressure exerted on the walking foot. (This is usually a knob at the top of the machine, but not all machines have this adjustment.)
Fabric stretches more along the crossgrain (grain running from selvedge to selvedge) than along the straight-of-grain (parallel to the selvedge). Not sure why yours is stretching opposite. Just wondering -- how do you know that the grainline that is stretching is parallel to the selvedge? Is it a directional fabric?