If you don't participate in a group celebration, some sort of acknowledgement about the birthday is still necessary.
Next time you see the person, have a card, or offer to foot the dinner or drink bill in honor of their birthday.
Go ice skating, perhaps, see a movie somewhere, or head to a Christmas market on a free afternoon.
You should have a general sense of who they are by now, and where things are heading.
This night is not about your burgeoning relationship.