You have 2x as many chances to get quads in Omaha however, you still need to have pocket pair. you can't start folding your full houses in Omaha because you are afraid of quads. It's just like in holdem and you have AA and AKK comes out on the flop. You 5 bet preflop so you opponent knows you have something good. Nothing you can do if he has KK and not AK, I'm never folding AAAKK on the flop. ever unless my opponent literally flipped his hand over. Then I might call praying I hit my ace on the river.
As far as the stats go I would say you have around a 4%-5% chance of getting quads in Omaha. You may not see them on holdem that as much because in reality, if I hit quad sixes, what is someone going to call 3 streets of betting with. Best case scenario would be J66 on the flop and he has AJ, turn is a blank and the river is a jack. Hopefully he puts you on a jack and is splitting the pot, but that's about the only way you are going to get quads all in there