After unlocking your Island Sanctuary in Final Fantasy XIV, you now have access to a very lucrative piece of side content. It's up to you to upgrade your sanctuary and learn what materials to keep in stock so you can benefit from the cowries your Handicrafts can generate.
Many of these handicrafts use Island Quartz, so you'll want to maintain a steady supply of this material so your mammets don't run out. In addition, recipes for a few mammet-sized tool upgrades needed to expand your island require this material.
Where to find Island Quartz in FFXIV's Island Sanctuary
Quartz Formation is the node to look for if you need Island Quartz. These nodes look like crystals sticking out of the ground, and you'll need an Islekeep's Bronze Beakaxe to gather any Island Quartz from them.

You can find these Quartz Formations around the following areas:
True Summit: The ideal place to gather Island Quarts is on the True Summit (X: 25.0, Y: 16.2). You need to be able to fly to reach this area, but it's worth the climb. There are six Quartz Formations on top of this peak, with just enough other nodes to make a fully functional gathering route. It's a small area that makes for efficient gathering.
Screengrab by Fanbyte via Square Enix Sunrise Lookout: There are only three Quartz Formations on the ledges around this area (X: 26.5, Y: 19.6), but with so many other nodes available, this is still an area worth considering if you need Island Quartz. This spot is perfect if you also need Island Iron Ore, which comes from the Rough Black Rock that makes up the majority of nodes in this area.
Screengrab by Fanbyte via Square Enix