They mostly tell you on these Safaris that you don't need a licence because it's off road, the age limit is suppose to start at 17 years old, but I have heard of kids being allowed on these quads.
Riding a quad is classed as a Hazardous Sport, which means you will not be covered by your Travel Insurance if you have an accident. The British Travel Insurance Companies can issue Hazardous Sports Policies which are suppose to cover Quads, Jeep Safaris, Paragliding, Horse Riding, White Water Rafting etc., but there are terms and conditions. Make sure you always wear an approved crash helmet.
The companies in Turkey you use will tell you they have Insurance Cover, if they do, it only covers Turkey, so if you miss a flight home through injury, you will pay for the next flight yourself.