While Square Enix intended for you to relax and just take things slowly after unlocking your Island Sanctuary in Final Fantasy XIV, there is plenty to be done inside this content. If you want to take advantage of everything your sanctuary has to offer, like providing a steady flow of gil, you'll need to upgrade your island and improve life on the sanctuary.
Island Beehive Chips and Island Apples are materials you might easily overlook since neither is found in many recipes. Beehive Chips are used in a single handicraft, while the Apples produce Island Sweetfeed and three handicrafts. Don't sleep on these materials, however, because they're easy to earn and can be exported to gain Islander's Cowries.
How to Get Island Beehive Chips and Apples in FFXIV's Island Sanctuary
Whenever you want to start gathering materials on your Island Sanctuary, you first need to enable "Gathering" mode. Doing so allows you to obtain materials whenever you interact with a node. The node you're hunting down for both Island Beehive Chips and Apples is the Apple Tree. For the Beehive Chips, You'll need to have upgraded your tools to the Islekeep's Iron Hatchet to gather the Beehive Chips, while Apples only require an Islekeep's Stone Hatchet.

The following locations are good gathering spots for Apple Tree nodes:
Eastward Stream: A great spot to gather Beehive Chips and Apples is around the river near X: 16.0, Y: 17.0. Counting the trees on both sides of the river, 10 Apple Trees are available. You'll only have to hit one additional node to get your first Apple Tree to respawn since gathering spots reappear after 10 other nodes were gathered.
Screengrab by Fanbyte via Square Enix The Wilds: A less-than-ideal but decent spot is slightly north of the previous one, in the middle of The Wilds. (X: 22.8, Y: 21.4). There are five trees in an almost straight line, though they are quite spread out. You will need to hit several other nodes to respawn these Apple Trees, but with other trees nearby, this can still be a valuable route to follow if you are looking for some wood as well.
Screengrab by Fanbyte via Square Enix