If you want to take part in tomorrow's Nozdormu Day event you best open up some space in your daily quest zone!
In patch 20.8 that dropped 2 weeks ago, Blizzard added a new quest that gives you 1500 XP on the rewards track for playing three games with Nozdormu the Eternal in your deck. This quest is going to be available monthly on the 15th, Nozdormu Day, and will take up a daily quest slot, in addition to your normal daily, (update: this IS your daily quest) so making sure you have room available is vital if you want to scoop up the XP boost.
This is the first month we'll be seeing the dedicated Nozdormu Day quest. With the XP reward, will you be taking part? Let us know in the comments what you are planning on playing with Nozdormu - you can share your decks with our deckbuilder.
The Hearthstone client doesn't support repeatable quests on a monthly schedule so implementing Nozdormu Day quests uses the regular quest system. Because of this, when you complete the Nozdormu Day quest that was given to you, it can never be completed again... so Blizzard duplicated the quest a dozen times.
If, in the future on Nozdormu Day, the quests don't activate, it is likely because of Blizzard not putting in a fresh round of these quests for us to complete. Game design is truly an art, and it's great to see they could leverage an existing system without new code to get this out quickly.
TL;DR: Hearthstone doesn't have a concept of repeatable quests outside the pool of dailies.