hello happy birthday

Month first is a typically US construction. "On April 9" and "on April 9th" are both common there. If you put the day first, you need to write "on the 9th of April". The "of" is vital, otherwise your message becomes confusing - "the ninth April" means nine years have passed! "The 9th" naturally implies "the ninth day".

Either April 9th or 9th of April can be used in speech. Written however, they are all accepted.

In conversation, 'on April 9th' is by far the most commonly used.