From Nookipedia, the Animal Crossing wiki
Palm trees, also called coconut palms or just palms, are a type of tree in the Animal Crossing series. These trees bear coconuts, a special type of fruit; however, they only produce two instead of the usual three fruits. Each coconut can be sold for 500 bells. Some beetles, such as the Goliath Beetle, will appear only on palm trees as well. In Animal Crossing: New Leaf, more fruit has been added, giving these trees the ability to bear bananas, as well as coconuts. In New Leaf, palm trees are found on Tortimer Island.
Palm trees can be grown by burying a fruit that grows palm trees near the beach, on the grass within ten spaces of the shore or on the sand. In Animal Crossing, coconuts can be obtained from the island, and in the other games of the series, coconuts will wash up on the shore about once a week. In Animal Crossing: New Leaf players can grow these trees from planting exotic fruit collected from Tortimer Island. The fruit that can be grown on the beaches are Bananas and Coconuts.