Welcome to the forums. Yes, the starter relay is behind the battery. Take an old screwdriver and jump the two posts on top of the relay and see if the starter will turn. If so, then the relay is likely bad. If the key is on, the ATV should start, but if the key is off, the starter should only turn. If the starter doesn't turn or the ATV doesn't start, test the starter.
You can test the starter by hooking it to a known good battery with a set of jumper cables. Hook one end to the battery. On the other end, hook the ground clamp to the engine or starter housing. Firmly touch the positive clamp to the post/terminal on the starter. If the starter is good, it will turn.
Most 4-wheeler problems are caused by a loose nut connecting the handlebars and the seat!!
You only need two tools in life -- WD-40 and duct tape. If it doesn't move and should, use the WD-40. If it shouldn't move and does, use the duct tape.