Palms start to grow on your Home Island as soon as you have reached level 27.
They will grow on the right side of the island, by the water. Please make sure you have enough free space there, so the palms can grow. Sometimes you need to cut the trees and grass, or remove rocks.

 

💡 Tip: explore expedition islands and adventure islands, some of them have tropic locations, where you stock up on palm logs!