First off she is two, Yep two. SHe doesn't have the mental capability to understand what she is riding or the risks YOU are putting her in. She doen't understand tire inflation, off camber riding, where do I put the gas, How do I start the thing, inclines, weight distribution and transfer and just generally don't do what dad does until you work your way up. Geez there many things that my oldest boy, 5 at the time had to know as part of the resposibility of running a thumb throttle, that even most 4 year olds wouldn't get. I've ridden with lots of them.
One of my inlaws is an excellent rider, takes huge jumps and skies it 40-50 feet easy. Well one weekend riding we were all gathered round watching him sky this thing over and over. About an hour after he was done there was some comotion over at the big jump... Well it seems that some little 5 year old cought up in the moment thought he could do that and while riding, parents watching him took off for the jump, full speed on a Honda 90. He skyed it, lost control, hit a tree and that time lived to tell about it. Parents got lucky. Could have had a tragic ending. These are some of the stupid decisions that 3-5 and some 6 year olds make over and over. Not because they are stupid mind you, but because they are immature.

Is she the only one with a quad?Probably not. So now she is riding with a different class of riders with 80's/125's/400's and such. SO now your two year old is now at the mercy of another 5, 6 12, 16 year old to make proper decisions EVERY time. So you kink it down right? Won't move with you on it so one day your playing on it and make a mental mistake and forget to kink it back down. She hops on it the next time and now your 2 year old is doing 17MPH, Faster than you can Run by the way, and wham into the street she goes. Get the picture? Hope the christ she is wearing the helmet that day......
Then there is the price... Electric quad $250.00 / Gas powered $1500.00+ and you have no clue if she'll enjoy riding one more than about 10 times and want to move onto horse riding.... Gonna get her a clydsdale? :-) Let her be a two year old, she has the rest of her life to ride ATV's.... But hey, It's your kid.
jarsong