No, under Georgian law, the surrogate cannot put any claim on the child. The birth certificate is issued with the names of the intended parents on it within 24 hours of delivery.
Read the full article on: Surrogacy in Georgia (Europe) – What Does the Law Say? ( 94).

Andrea Rodrigo
Bachelor's Degree in Biotechnology from the Polytechnic University of Valencia. Master's Degree in Biotechnology of Human Assisted Reproduction from the University of Valencia along with the Valencian Infertility Institute (IVI). Postgraduate course in Medical Genetics.