Buzz implies interference (and it would be easier to figure out what if you could tell us the frequency - although if it is very high pitched I'd assume a faulty or misconfigured digital effect)
Generally speaking, effects pedals should not introduce buzz unless either they are faulty, or in the case of digital effects you can sometimes find settings which will introduce unwanted frequencies. A multi-effects pedal should be no more likely to introduce noise than a single effects pedal.
It is possible that you have this buzz on an undistorted guitar, but the distortion you use is amplifying it, as zameb commented.
One thing that can help is to use a noise gate before your distortion (I think the G1On has one in the Dyn/Fltr section before the OD/Dist section) - this will reduce/remove noise when the guitar is silent.