Should I send this boy a birthday message?

Sure, I want to send it. Wish him good luck from an ordinary friend's point of view and create an ambiguous atmosphere.

If he is not interested in you, then he will only take it as a blessing from a polite friend, and he will pay more attention to you and appreciate you; If he is interested in you, that would be great. This short message is an opportunity, and you have started to develop slowly. You must not miss this opportunity.