if the tip of your needle is breaking, it's because the needle is not coming completely out of the fabric before you move it. Because the needle is not out of the fabric, you bend it a tiny bit when you move the fabric and the needle hits the plate and breaks.
It's either because you are moving the fabric too quickly or the foot you are using is not applying enough downward pressure to allow the needle to get entirely out of the fabric at the completion of the stitch. It's letting the fabric move up when the needle raises, so it's still stuck in there. What kind of foot are you using? A hopping foot is great because it holds the fabric down for the stitch then hops up, releasing the pressure on the fabric allowing you to move it. If you don't use a hopping foot, try slowing down a bit when moving the fabric or try adjusting your presser foot pressure down a tiny bit. FMQ does take a LOT of practice to get your machine speed and hand speed to be coordinated. This is one area where taking a class is really helpful.