no, you can get quad bikes for regular road use, not just farmer quads.
the basic rules are the same as motorbikes, you must be taxed, insured and have a bike license etc, the main difference is there is no legal requirement to wear a helmet.
as it is classed as a bike, number plate only needed on the rear
quads without a number plate are illegal but hard to catch as they are narrow enough to dive through woods / into fields to evade police vehicles.