"It's your birthday in June" "It's my birthday on September 6th" IN a period of time, or ON a specific date.

The correct way to say it is, "My birthday is in June." "It's my birthday in June." is correct also but since this sounds like a response, you should emphasize the "is" like this, "It is my birthday in June."

IN JUNE

Your birthday is in June! But if you want say the day, you must use ON, for example: my birthday is on 17th of June.

in June