So I can never really pinpoint an answer, but I can reduce a few things.

1. So my first thought of when his Birthday could be is late May to early June because he is a part of the Gemini coven. This is obviously not true because by May 10th he was already 22.

2. Because he was born in 1972 his birthday 1994 is the year he turned 22 so his birthday is anywhere from January 1st-May 8th. Jo or Kai would have mentioned if the events of May 9th were on their Birthday.

3. Gemini merge is on or around their 22nd birthday and the news of him not being able to merge seemed to be new information by his reaction. So I believe his birthday had to have just recently passed from May 9th. Which reduces the range to April 1st-May 8th.

4. He has traits more suited for Aries than Taurus based on this websites description: 

Which isn’t all that concrete because that doesn’t describe him that much but it describes him more than Taurus.

5. Contrastly Jo has traits closer to a Taurus and since their twins they share the same birthday. Unless he was born at 11 something and she was born just after midnight. Unlikely but cool if it was true.

6. Those last two points are all speculation based on my vague knowledge of signs. However that site had a very accurate description of Bonnie:

Which is why I trust it. 

7. I wish they would have panned to his ID in 6×09 so I wouldn’t have to do this guessing game, but it is a minor detail that it took years to get that info from the main characters. 

Side note: Season 3 we find out when both Elena and Caroline’s birthdays are, we don’t find out Bonnie’s until season 6. Even sadder, they don’t celebrate her birthday until the one year she’s not around to celebrate it.