Not sure of the local requirements but check with your travel insurers some don't cover quad bikes.

Make sure any vehicle you hire is in good condition and check that you’re insured. When renting mopeds or quad bikes, insurance sold by the hire company usually only provides third party insurance, which only covers the cost of damage to another vehicle. Quad biking is considered an extreme sport and carries the risk of serious injury or death. Specific travel insurance to cover quad bike rental is essential to avoid you having to pay the costs of private health care and/or repatriation to the UK. Always take care to read the details of your insurance cover before you travel on holidays paying particular attention to the small print and exclusions on your insurance policy.Greece has recently introduced legislation banning the use of quad bikes of 125cc and under on tarmac roads. A fine of €1000 can be imposed. Larger quad bikes (150 and 310cc) are available to rent, but only to drivers aged 23 and over with a full driving licence. Provisional licences are not considered adequate for driving in Greece and can invalidate your insurance.If you intend to hire a moped you will need a valid driving licence with at least category A1 - ‘light motorcycle’. Category P, which is valid in the UK for driving mopeds up to 50cc, is not valid in Greece.By law you must wear a crash helmet on a scooter, moped or motorcycle. Quad bike riders must wear a full-face helmet (or non-full-face helmet plus goggles). Failure to wear a helmet might invalidate your travel insurance if you are involved in an accident and could lead to a €350 fine and confiscation of your licence.

https://www.gov.uk/foreign-travel-advice/greece/safety-and-security