Depends on the size of the quad..
It is called a heavy cycle believe it or not and therefore a quad can be driven on a bike licence.
My son is very heavily into the quad scene and would have one on the road tomorrow if he could get reasonable insurance ( theres that word again ! ), but although his R6 insurance was ?220, the cheapest quote he could get for any sort of quad was over ?500.
We even looked into getting it done as an agricultural vehicle.. that was possible but still the insurance was the same c7u8