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!
You must be a minimum of level 78
You must have completed the quest "Nordrilund Exposure"
You must have completed your school’s level 58 spell/pet quest from your professors.
Once you meet all of these requirements, then you can go talk to your professor about getting a new pet!
You have not completed the quest "Nordilund Exposure" - continue your tasks in Grizzleheim and Wintertusk by completing the quest you currently have called "The Bigger They Are" that was given to you by Valgard Goldenblade in the Northguard Throne Room. That will open other quests in Grizzleheim and Wintertusk that will lead to unlocking Nordrilund Exposure