I doubt it, but I don't know for sure. You need to kill so many things just to gauge whether the bar progresses... But I imagine it's just like normal weapon leveling, so it probably won't count. Easiest way to level it; just do a bunch of low level moon quests, like Dark Beyond and Shrine of Oryx (my personal favorite). Even for a common, it performs pretty darn nicely once you get used to that recoil.