During some quest chains like
https://www.wowhead.com/achievement=13542/how-to-train-your-direhorn
https://www.wowhead.com/achievement=13030/how-to-ptrain-your-pterrordax
or the Broken Shore treasure hunt, you often have to wait a certain timeframe (i assume it is mainly until the next daily quest reset) until the next quest is offered.

Would it make sense to keep this kind of info in the db?

Normally we attempt to do things like that. For example, dailies have a reset time and Blizzard API usually indicates that the quest is available once again. For quests that are only available during holidays we use calendar API to determine whether we are in those holidays. If there are ways we can determine the rules of when a quest should be available we can attempt to model that with information we can get from API.

The direhorm and pterrordax chaoons have a second set of conditions , you have multiple sets of dailies that increase the reputation bar above the npc and once you level up, the next set or the next part of the chain become available.

https://www.wowhead.com/faction=2233/dino-training-pterrodax#objective-of

https://www.wowhead.com/faction=2370/dino-training-direhorn

The legion quest has no such thing.