The names of these islands have changed. The Tempest islands that this quest targets are in the center of the lower abyss, Sullenscale Rive. You must visit them starting with Ulna Isle and continue clockwise. You can see the island names by opening your map and zooming in. You have to fly low to get credit for each island, but you don't have to go within aggro range. Most of the time, the spot to trigger the quest update is near the center of the island. You can make this quest much easier by going to Redemption Landing in the upper abyss first and grabbing the Wing Recovery buff.

The order is:

Island of Waves (Ulna Isle)
Zephyr Island (Hamate Isle)
Island of Clouds (Capitate Isle)
Gale Island (Pisiform Isle)
Island of Cold (Lunate Isle)
Storm Island (Carpus Isle)
Island of Wind (Scaphoid Isle)
Island of Gust (Triquetral Isle)