I have never come across academic transcripts containing date of births ?
However , Any time you provide conflicting or contradictory information to CIC
you do run the risk of it being picked up and being identified as someone who has provided fraudulent information and it may have more serious repercussions.
If you dont want to take a gamble , I would suggest that you ask your friend to contact the University and try to get them to correct it to what is shown on the passport.