Learn English – Correct way to ask about age

You do not need to add "can", However, your construction of "details of your flight" is awkward. I would recommend one of the following:

"I was wondering if you could give me your flight details..."

or

"I was wondering if you could give me more details about your flight..."

"How old is your unborn baby?" is not grammatically wrong, but is not the way people usually ask the question. We normally count age from birth, not conception -- I presume because we know exactly when a baby is born, but we rarely know for sure when he was conceived -- so to be consistent an unborn baby's age would be a negative number!

The common way to ask the question is, "How far along are you?" or "How far along is your baby?" A woman will often say, "I am three months pregnant". But then we also say "the baby is three months old" or whatever number.

Doctors will say that a baby is "three months gestation". Sometimes they talk about age since the mother's last menstrual period, as in, "the baby is three months LMP". LMP is also a date that the mother is likely to know. It's usually a couple of weeks before conception, but close enough. Medical researchers talk about "embryonic age", which may be abbreviated to simply "age". But these terms are rarely used in casual conversation.