If you give birth while in Greece you must provide your new born baby with a birth certificate. Generally it is advisable to get help from an organisation or volunteers in your camp with that. In hospital you will get a birth declaration issued by the doctor or the midwife that participated to the birth, that contains all the relevant information about the parents and the new born baby.
Up to ten days after the birth, you must go to the Registry Office (ληξιαρχείο) of the municipality where you reside and register your new born baby. You must bring the birth declaration from the hospital and all the documents that state the family status. They should be translated into Greek but in practice the Office seems to accept them in the original language, as long as the parents can declare the content of the documents. If one of the parents cannot go to the Registry Office in person, the other parent should bring an authorization from the missing partner. It is always better having a Greek interpreter with you, since a translator mostly is not available at the Registry Office.