1. Look for "acronyms" in the search function.
2. FMQ is the quilting that is done by dropping the feed dogs on your machine and moving the quilt sandwich by hand. In other words, the quilter controls the motion of the quilt. They key is to find the sweet spot between the speed of the footpedal and the speed of the motion.
3. A fat quarter is the equivalent of a 1/4 yard of fabric. But the fabric is cut at the 1/2 yard mark and then cut again in half crosswise. This gives you a larger piece than the skinny 1/4 yard and there are books written for the optimal use of FQs.
4. A LA is a quilting machine that has a really large throat and it is used for quilting. LAQ is the Long Arm Quilter who hires her (or his) skill and machine. These machines are quite pricey and they take up quite a bit of space. If you want to check out LA work - look at Charisma's posts. Her work is amazing.