Patch of green space on island where nothing grows

Most Island Peninsulas aren't that big, they may seem so, but they vary rarely go over 3x3 (a tree needs 3x3). They round out and add some extra land, though not always a full grid space. so you can likely fit two trees on one at best. Most players I've come across use this peninsula for furniture arrangements instead, like Lighthouses or small picnic areas. It seems to be Designed/intended for that.

As Rapitor asserts, it is unlikely to be a problem with the land per-se, and much more likely an issue of hole area.

One way to confirm whether or not a space is valid for growing trees is to... try and plant a fully grown tree there!

Simply chomp a fruit, add a nearby tree to your pockets, then approach your target hole and attempt to "plant" the tree from your inventory.

If the hole is in a valid position, your villager will insert the fully grown tree into the spot. If the hole is invalid for planting, your character will say as much.