Hi,
Assuming the quad has a normal ignition setup like all bikes, you have obviously got a problem with the ignition switch which you should be able to diagnose with a continuity test meter.
You should also have a kill switch on the handlebar which works separately from the ignition switch. If you select the handlebar switch to "stop" while the engine is running the engine will stop, if you tried to start the engine with the kill switch selected the ignition lights come on but the bike won't run.
It is highly unlikely to have a defective ignition and kill switch, so my money is on the ignition switch.
Cheers
westcountry
December 2013