몇월인가요 is composed of 몇월이(다) + ㄴ가 + 요 몇월이다 means "be what month" ㄴ가 is an interrogative ending 요 is an honorific ending ex) 그게 누군가? Who is it? 그녀는 예쁜가? Is she pretty? 저게 학교인가? Is that a school?

몇월인가요? 는 what month is it?

To make an interrogative sentence, you could use 인가요? 입니까? 이죠? at end of the it. 인 itself doesn't mean anything.

verb 이 is similar