Was reading the latest Visitors Guide from CYTA and it mentions a Cyprus Climbing and Mountaineering Federation. Their website appears to be
http://www.komoa.com although it is heavily under construction and mostly in Greek. The contact page does have the number of someone you can contact though.
The guide also says "Several locations in Cyprus are suitable for rock climbing, such as Drousia and the Valley of the Diarizos River in the Pafos district, Cavo Greko between Protaras and Agia Napa, and Trooditissa's excursion area in the Troodos Mountains. There you can do almost everything: traditional climbing, sport (bolted routes) and bouldering."