My parents have a place in that area, and there looks to be plenty of rocky trails that could be ridden with some reasonable hills and stunning scenery. However, I’ve not actually gone biking over there, and I’ve not seen a bike shop in any of the nearby towns. The Tour of Turkey road race usually goes through Bodrum, so there must be some interest in bikes generally, but I’ve not managed to find anything.
Apparently the guys who work at the Neilson in Ortakent know a few of the trails around there so they may be worth speaking to if you’re in that area.
I also have no idea of the access laws and using the trails, but the guys above should hopefully be able to help with that. Finally, maps seem somewhat limited, I was unable to find anything large scale in any bookshops or other places that might have material, but again, I haven’t spent hour searching.
Good luck if you do go and get to ride, it would be interesting hearing back how you found it.