I would definitely switch to a better quality, stronger thread. Do you still have some of the thread? If so, hold a piece in both hands and "snap" the thread. Weak thread will break easily. Weak thread is not only associated with poor quality thread, but also with age of thread. It's possible that the thread you bought was old and not stored properly, or even a bad manufacturing batch.
However, I am also wondering if the quilting was done with tensions too tight on the bobbin and top. When you quilt, you may need to loosen both tensions so the straight stitches have some "give". An easy solution to this for you would be to change from a straight stitch to a narrow zigzag stitch when you stitch in the ditch. The zigzag provides some "give" in the stitching line. The snapping of threads you are talking about in a quilt is the same as what happens when you sew stretchy fabrics (such as stretchy knits) with a straight stitch; the fabric has more "give" in it than the stitching line has, so the thread snaps.
Also, what brand and type of batting did you use? And how long are your quilting stitches? Seems to me a high loft batting and a long stitch could combine to put extra stress on the thread, especially if the bobbin and top tensions are too tight.
Edit: I just want to add that you typically do not want to use "quilting" thread in a sewing machine. Usually "quilting" thread is glazed and is meant for hand quilting. However, this kind of thread does not explain the thread breakage you are getting. Glazed thread does mess up your sewing machine, though.