That may be an overflow, but also a drain line for the float bowl. Check the tightness of the drain screw at the bottom of the bowl. Do not overtighten, as the carb metal is pretty soft, and over tightening can cause leaks itself. Which of course sounds like your problem.
Perhaps your float bowl level is set just a bit high, which would be noted as a tendency for the quad to stall out from an overly rich mixture (usually when the engine is hot) and even flooding out. Check for black smoke on acceleration. If the quad is brand new, take it back to the dealer and have them reset the float bowl level. If not, get a shop manual and reset it yourself. Not too difficult a job, but I will not go into it here, as having the procedure in front of you helps get it right the first time. Following detailed instructions on a forum can have unsatisfactory results, as the advisor can sometimes forget a step, or leave out important info.
2003 Rincon 650 w/mods
2003 Rancher 350 w/mods