Are you on visa or eTA to Canada? If visa - not an issue. If eTA - they will not get the authorization from Ottawa if there is discrepancy between DOB on eTA and in their system. The information is stored in the reservation under DOCS SSR (sounds Chinese to you? Just google it out - here is an example with IATA mandated format: https://guides.developer.iata.org/v212/docs/docs). The good thing is that it can be changed in the system without any issues . It can be done even at OLCI or agent check-in. Plus normally TK swipes passport at check-in and the swipe data trumps any info previously stored in the reservation.

So normally, I would not worry too much. If 'computer says no' ask them to swipe passport first and then request eTA authorization - you should be good to go then.