Fire Ammonite

If you go out of the front of your garrison and look to the right you'll see a lake of lava. Start fishing here and within a few casts you should catch a [item=110291]. Use this item and you will now be able to catch Fire Ammonite from the lava. Once you collect ten Fire Ammonite you need to right click the stack which should give you something like 5-7 [item=112628]'s needed for the quest. I caught 20 Fire Ammonite and was able to get 12 eggs from them.

The larger lava lake north northeast of the garrison did not yield any fire ammonite / fire ammonite bait for me. The small lake right next to it worked fine. This is probably related to the large lake being part of the garrison area, while the small one is part of the Frostfire Ridge area.

You have to fish up: 20 Small Fire Ammonite or 10 Fire Ammonite or 5 Enormous Fire Ammonite and click the stack to have Fire Ammonite Egg.
You can fish Fire Ammonite in any lava pool into the Frostfire Ridge map and/or into your Garrison if you have the Fihing Shack building (You can fish it into the lava pool and/or into the normal one).
It's better to have an high fishing skill, use a Fire Ammonite Bait and a lure to catch more quest items in less time.
You can also kill Pale Fishmonger as Perculia wrote to drop some Fire Ammonite. Watch out that he drops Fire Ammonite just the first time you kill him!