Difference between 月 and 月份 for the noun "A month"
Both 重要的月份 and 重要的一個月 works fine for translating "an important month". But 重要的月 minus the unit of measurement (份/一個) sounds more like you're speaking of the Moon.

The difference between 月 and 月份 is that the latter is specifying a month of the year, while the former is more ambiguous: it could refer to the moon, or a month's time. Thus 月份 could be used standalone but you would normally prefix 一個 to 月, as you did.

It is perfectly okay to say 斋月是伊斯兰最重要的一个月.



I suspect the reason your friend corrected you was because the topic was Ramadan.

Traditionally (in some parts), 九月 means September in the Gregorian calendar, while 九月份 refers to the ninth month of the traditional lunisolar Han Calendar. By extension, he might have felt the ninth month of the Islamic Hijri Calendar (which is what Ramadan is) should also be a 月份.

This isn't a matter of grammar, but rather a customary usage.