Disclaimer: I have yet to be very successful with FMQ. So this is just ideas from what I have read. The bigger needle suggested may help. Or just try a new needle.
Thread breakage can be caused by the thread being abraided on some edge. It may touch an edge due to the motion of moving the quilt will pull the thread to a side of the plate that it doesn't on normal sewing.
So i would start with a new needle, then if still a problem, look at the hole in you needle plate and around your bobbin case to see if there may be a small burr or something that may fray the thread.
Good luck!