Out With the Old (Alliance)

Quest showed up near Fort Victory in Nazmir. The stone is called the Shrine of Dawning and gives this quest. I cut 4 insightful Rubellites and was rewarded 2g, 34s. No follow up quest, nothing. EDIT: Once you do this quest the stone will be viewable on your map and you can return (once per day?) to harvest gems from it. Once you click it the gems will appear in a basket just in front of the stone. I believe you need to have completed the Jewelhammer's Focus questline.

From the Nazmir stone I received:
1x Solstone
1x Kubiline
4x Rubellite
1x Amberblaze
1x Tidal Amethyst
7x Scarlet Diamond

After creating a [item=153716], jewelcrafters can pick up free gems every day from of six shrines, each corresponding to a gem colour.

Only shrines that have been unlocked (by completing a quest) show on the map, and can be interacted with.
Some quests may have already been completed during the previous questlines, if they're not showing you can confirm they're complete with the following macros:

Tiragarde Sound, Blue Gems - Tol Dagor: Jewel of the Tides, Tol Dagor dungeon, completed during jewelcrafting questline
/run local id=49575 print(("The quest with ID %s is %scomplete."):format(id, IsQuestFlaggedCompleted(id) and "" or "not "))

Drustvar, Purple Gems - High Prospects, completed during jewelcrafting questline
/run local id=49576 print(("The quest with ID %s is %scomplete."):format(id, IsQuestFlaggedCompleted(id) and "" or "not "))

Stormsong Valley, Yellow Gems Cracking the Surface, completed during jewelcrafting questline
/run local id=49577 print(("The quest with ID %s is %scomplete."):format(id, IsQuestFlaggedCompleted(id) and "" or "not "))

Zuldazar, Green Gems - Atal'Dazar: Not All That Glitters..., Atal'Dazar dungeon
/run local id=55582 print(("The quest with ID %s is %scomplete."):format(id, IsQuestFlaggedCompleted(id) and "" or "not "))

Nazmir, Red Gems 4 - Out With the Old, Hand in 4 Insightful Rubellite
/run local id=55583 print(("The quest with ID %s is %scomplete."):format(id, IsQuestFlaggedCompleted(id) and "" or "not "))

Vol'dun, Orange Gems - Sun-Speckled Dunes, Collect gems and polish as per DMF glass quest
/run local id=55581 print(("The quest with ID %s is %scomplete."):format(id, IsQuestFlaggedCompleted(id) and "" or "not "))

Corrected coordinates for TomTom / Paste users are:
/way Tiragarde Sound 46.36 23.45 Blue Gems - Shrine already unlocked
/way Drustvar 34.13 35.46 Purple Gems - Shrine already unlocked
/way Stormsong Valley 60.70 58.51 Yellow Gems
/way Zuldazar 43.14 64.35 Green Gems - Atal'Dazar dungeon to unlock
/way Nazmir 61.32 37.24 Red Gems 4 - Insightful Rubellites to unlock
/way Vol'dun 44.18 38.05 Orange Gems - Collect gems and polish as per DMF glass quest to unlock.

For reference: You need to have done the JC questline ([storyline=887]) that unlocks [item=153716], before this quest becomes available. However, you don't need to have [item=153716] crafted to do the quest. You do need to craft it to access the daily gems from the shrine though.