Is there a "Nirnroot" quest?

Solution 1:

There is indeed a Nirnroot quest:

When in Blackreach (the underground city), you'll find a quest called "A Return to Your Roots" which will task you with finding out the significance of the crimson nirnroot as well as send you on a collection quest to get a bunch of them.

Additionally, Ingun Black-Briar in Riften asks for 20 of them as part of a miscellaneous objective.

Outside of these two quests, nirnroots are used in alchemy to craft invisibility potions.

Solution 2:

There's also someone in Riften who asks you to collect a list of alchemical ingredients for them, 20 Nirnroot is part of the list.

Solution 3:

The Crimson Nirnroot in Blackreach is part of a quest. You start the quest with the discovery of the first Crimson Nirnroot. Collect as many as you can, and when you find Sinderion's Field Laboratory, you will find Sinderion inside. He will have a book clutched in his skeletal hands which allows you to start the second part of the quest.

You will then be required to collect 30 Nirnroots. It's a menial task, but they aren't hard to find and currently I have found 37 of them. Good luck.