I think that your source must have transcribed the reading wrong. The readings for 月 are: つき (tsuki) when standalone speaking about the moon がつ (gatsu) when in reference to the months (January, etc.) げつ (getsu) when counting months as a time frame 二ヶ月 (nika/getsu, two months). I hope that helps!

Every month 1 to 12 you have to say "gatsu". ichi-gatsu ni-gatsu san-gatsu : : : There is actually no "katsu" reading for 月. There are "getsu" or "gatsu" besides "tsuki" for example 　夕月　（yuu getsu） 　年月　（nen getsu） -> time 　正月　（shou gatsu） -> new year 　一ヶ月　(ikka getsu) -> one month = for all numbers But if you learn "Japanese" month of the year, it will be more difficult, junior high school students would learn it at school. Mutsuki = Jan Kisaragi = Feb Yayoi = Mar Uzuki = Apr Satsuki = May : : I wish that you did not mention it.