Those happen to everybody.
It may be your tension, it may be your speed (you need to find the right hand speed/needle speed ratio). You might even need to just go slower on curves, but you are good on straight-aways.
However, it could also be your needle (so put in a fresh one), or your fabric. I'm a somewhat experienced FMQer and got eyelashing on my last project. It only occurred on ONE fabric of a patchwork piece. The rest of the quilt everything was perfect. I ended up having to go really really slow, and ripping out a ton. Changing to a new needle helped a little bit, but it was a horrible pain. I had to turn the quilt over to check every section over this one fabric. I hadn't had any eyelashing like that in like 6-months, so I was not happy!
Good luck!