Lost In The Deeps

After you get through the colossal worm's loop and into the first main chamber, take the first right and then an immediate left. Pebble will be channeling some green tendrils. Then just backtrack to get out.

When you get to the colossal worm's loop, go over the rocks in the middle part where there is a sort of ramp. When you come over the other side of those rocks, be careful not to fall or Pebble will disappear. And of course, be careful not to get eaten!

If you lose Pebble, go out of the cave until it says "Deepholm Scoured Reaches" and go back in. He won't respawn otherwise.

It's probably one of a random selection of dailies, so you might have to wait a day or two before it's part of your selection.

The quest is given by one of the smaller Therazane in the throne room.

EDIT:
Yesterday I got the quest from Pyrite Stonetender. But today he doesn't have any quests for me. :(

EDIT2:
I got another quest that wasn't available to me yesterday: Underground Economy.
So it does seem to be part of a random selection.

EDIT3:
Confirmed breakdown of Therazane Dailies (with associated achievements):
Fixed Dailies
Fungal Fury (Fungal Frenzy)
Soft Rock
Through Persistence

Fear of Boring
Motes

Plus one random from the following
Beneath the Surface
Lost In The Deeps (Rock Lover)
Underground Economy

At Revered rep, another random from the following is added
Glop, Son of Glop (Fungalophobia, The Glop Family Line)
The Restless Brood (My Very Own Broodmother)

PS: (update after 14 years 😂)
It's very easy to get Revered with [spell=78635]. I hit Revered after completing the first day's worth of dailies, and immediately picked up the Revered daily.
Note that in Cataclysm Classic, everyone gets Mr Popularity, even if not in a guild.

Dammit, Corki!