"It's your birthday in June"
"It's my birthday on September 6th"
IN a period of time, or ON a specific date.
2014年10月7日
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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."
2014年10月7日
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IN JUNE
2021年6月29日
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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.