Ok, this might sound stupid, but when I first started FMQ, I kept forgetting to put the presser foot down. Because it doesn't grab the fabric as hard as a normal presser foot, I couldn't distinguish as easily between it's up and down position, and kept forgetting to put it down. That led to a lot of problems, including broken needles. I still feel pretty silly about it, but I can't be the only one who has ever done that (I hope!)
Also, make sure you're using the correct type of needle for your machine and inserting it properly.