History on this is a little sparse, but what I can definitely say is that as long as Genesis has been written there was celebrations of birthdays. 40:20-20 states:

Now the third day was Pharaoh's birthday, and he gave a feast for all his officials. He lifted up the heads of the chief cupbearer and the chief baker in the presence of his officials. Source.

The guess is around 2,500 BCE when birthday celebrations started (Source). Cake was later introduced by the Greeks as a tribute to Artemis in a moon-shaped form. Source.

We can thank the Romans for taking the idea further and celebrating birthdays with common man (same sourec as previous).