Is it permissible for a grandmother to receive a retirement pension from the Turkish state, given that she holds two nationalities and lives in Syria, and is it permissible for her to receive retroactive payments?

The permissibility of taking money is subject to the laws regulating this matter. It is necessary to understand and abide by the conditions for entitlement to a pension, based on the hadith: "Muslims are bound by their conditions." The question regarding the entitlement of someone holding two nationalities and living outside the country should be directed to the relevant authority. If it is permitted, then your grandmother is entitled to the pension; otherwise, she is not.