Plenty of flying schools around (Pahos/Larnaca) but I don't think you mean 'flying lessons' I think you are looking for a flight in a light aircraft and a chance to have a go at the controls. You need more than just a birthday present if you are going to go for a PPL.......that is unless the person giving you the present of flying lessons has very deep pockets!!!!!!
I took a Piper Cherokee from Paphos a few years ago with Griffon Aviation but although I have a PPL, I still needed a check pilot with me. although I did all the flying. Myself and a neighbour went up for about 1 1/2 hrs and then (2002-3) it was about CY£75, which was cheap as far as comparrisons with the UK go. But I suspect it will be at least double that now.
But try it......it's great fun.