Γεια σου Ιάκοβε
You can ask: "Πόσων μηνών είναι αυτή;"
or "Πόσων μηνών είναι αυτό;" They are both the same correct.
In the question "Πόσων μηνών είναι αυτή;" (= How many months old is she?), αυτή corresponds to the word "η κόρη" = daughter (feminine).
In the question "Πόσων μηνών είναι αυτό;" (= How many months old is it?), αυτό corresponds to the word "το παιδί" (= the child) or "το κορίτσι" (= the girl) or "το μωρό" (= the baby), which are all neuter.
However, when the baby is next to us and we are talking about it already, it sounds much more natural in Greek to say "Πόσων μηνών είναι;" .In Greek we omit the personal pronoun when it is obvious who we are talking about.