Is it possible for me as an developer/owner of my paid app (released on the Google Play market) to unlock its full version for specific user?
It's been discussed on the Google forums. Basically, there isn't a really easy option, but the options discussed in the thread are:
The quickest and easiest of the two is to use the gift cards. But the most suitable for long term is the unlock codes. An additional benefit of the unlock codes is having the infrastructure in place for any future apps you develop. (Or maybe selling the library to other developers).