There are 3 requirements to meet to be able to accept the starting pet quests from your school professors in Ravenwood, Nightside, or Krokosphinx. Each school has a different quest and each school has to fight a different boss in Wintertusk to get their new pets!
You must be a minimum of level 58
You must have completed your school’s level 48 spell/pet quest from your professors.
You must have completed the quest "Nordrilund Exposure" given by Grandmother Raven in Hrundle Fjord.
Once you meet all of these requirements, then you can go talk to your professor about getting a new pet! Talk to your same professors who you received your level 48 spell/pet quest from to get the quest.
I'm guessing, without auditing your account, that you do not meet the third requirement. If you haven't started already, you may want to get caught up on your Grizzleheim and Wintertusk quests.