Simple answer is do not hire one. There are lots of tales of woe in the past on HT of people who ignored the advice.

Legally you require a FULL driving licence but reputable insurance companies will not cover you in the case of an accident

That's correct! It's better not to hire a quad bike in Greece at all. Many are poorly maintained and they are not the safes vehicles to being with.

I recommend hiring a convertible. Get yourself ready about driving in corfu with this guide and enjoy the mobility, freedom, fresh air in a convertible. You will feel 100% safer and it's just as much enjoyable as riding a quad.

Honestly I think exploring Corfu in either a Fiat 500 Convertible or a Mini Convertible is the best way. You'll have some fond memories to take home.