First you need to know her registry details. You can get a "detailed" copy of "your" registration (also showing your 1st degree relatives with names, their parent's names, birth places, dates etc.) by applying to ANY Civil Registry Office in Turkey. Then you have the apply to THE Office of the county you're registered at. And this only first degree relatives. For a grandmother you may try your chances and ask a friendly help from the Civil Registry Officer, if unsuccesful you'll have to go through a court process.
As suggested in a previous answer it may be a good idea to get some help from a local researcher who could assist you with this as they may have connections and access to some data. I think there is a resident researcher on the APG list now.