I would try a different thread. I had a really pretty varigated King Tut thread that was so linty that it made a little lint ball at the needle. I snapped two needles before I finally figured out that the little lint ball was acting like a knot right at the needle. I changed out my thread and problem solved.
Do you have the larger zig zag hole plate on the machine while FMQ? If you are pulling the quilt sandwich enough to move the needle, it can skim the hole plate causing the thread to spread and break. Try to have the quilt all up on the table so there is no drag on the needle.