The best figs are available from mid-september and, provided the locals haven't harvested them already, you'll find them in the river valley between Hiroshima and Kaya and also on the headland surounding Af Kule. There are also 3 or 4 on the old greek road between Kaya and Fethiye.
While you're out searching for figs also keep an eye out for carob or locust bean trees. Even if you're not keen on them, they're a great free, non-toxic treat for dogs.