No, 'a month of the year' can only be en månad på året. In the plural, årets månader is good for 'the months of the year' though.
If you add something, you get more possibilities. Like, 'the second month of the year' can be both årets andra månad and den andra månaden på året. But just on its own, årets månad sounds very odd, and årets månaden is never possible because you can't have a definite form owned by a noun in the genitive. (that would be like saying the year's the month in English).