The birth of the GC is not like the birth of ''ordinary" babies. You will see a popup message informing you of the event, and the villagers will gather in Village Center to celebrate the occasion. In addition, the GC is 5 years old when it is born, and is not cared for by its mother. It's unlikely that you missed it.

After you finish your current game, you might want to download and install v1.1, which was released just before Christmas. Be sure that you are ready to start over, though, because the savegames are incompatible and it will delete any v1.0 savegames that it finds on your device.