The bible does establish Birthdays as a well known celebration that pagans did.
Jews like Herod that were heavily influenced by Pagan practisies happily celebrated birthdays. (The Jewish calender is a very complicated one and actually has many more days like febuary 29th in our calender so doesn't lend itself to celebrating birthdays easily.
So did both Jews and christians including Jesus (who became aware that His cousin was murdered as a birthday event for Herod) suddenly turn around and start celebrating birthdays?
No! neither Jews (unless pagan influenced ones like Herod) or Early Christians although knowing about birthdays celebrated them.
Why would I as a christian celebrate something that Jesus and the other christians didn't when they easily could have? and thats the point. Birthdays were known but Christians simply didn't do them including Jesus himself.
Do you have any strong religious convictions you would like your wife to respect if nothing else? Wouldn't it be better for you both to come to a mutual agreement, maybe have special dinners out as a couple without it needing to be on a particular date like birthdays or Christimas etc that will spoil the evening making it a battlefield? I know if had a wife with different religious convictions I would respect her beliefs and allow for them as part of my love for her while not compromising my own.