You will be able to harvest Gold Coin Fruit at the end of Chapter 5 when you defeat the boss, Eldest of the "Black Rabbit Brotherhood". After beating the boss, the building in the front will open, and there will be an elevator in the back. It will take you up to the "Lost Flower Garden, which is a secret part of Hotel Krat that houses the tree that creates Gold Coin Fruit.
You will find the NPC Giangio that you first encountered in Chapter 4, in "St. Frangelico Cathedral Chapel". He will be a vendor who will sell you Cube Wishstones in exchange for Gold Coin Fruit. Interact with the tree and collect your first Gold Coin Fruit, and the achievement will unlock!
Quest line involving Gold Coin Fruit, by Isomirism - "You also need a Gold Coin Fruit if you're doing the red fox questline in chapter 7. He will give you an emote "beg" and a record "Quixotic" if you give him a Gold Coin Fruit at this moment." There is an achievement for obtaining all emotes and records. Additionally, in Chapter 11, you can skip the boss fight against Black Cat (the brother) if you give him a Gold Coin Fruit. Although the fight is very easy, and you get his Mask when you win.
Most importantly, you will need Gold Coin Fruits to Respec your Level (returns all Argo), P-Organs (returns all Quarts), and Legion Arm. Each time you Respec, the cost of Gold Coin Fruit will increase! You will be able to Respec at the end of Chapter 7, after defeating the boss "Champion Victor", which will give you the key from Simon Manus to unlock the door that houses the "Saintess of Mercy Statue".
In the 1.2 Update (09/27/23) - You can now Respec using the Gold Coin Tree!
Location of "Saintess of Mercy Statue" to Respec - The Respec room was actually on the way to the boss, it was the large locked door, before the large area with the palm trees and a huge buff enemy in the center, but I will explain how to get there from the boss room. Firstly, exit the boss room from the entrance, go up the stairs on the right, and go down the stairs to the large area, turn right, and straight ahead down the short staircase is the large door where the Respec room. Inside there is also a Stargazer for fast travel named "Saintess of Mercy Statue".
The tree will create 1 Gold Coin Fruit every 10 minutes in-game, and the maximum capacity is 8. But there are several upgrades and items that will improve/speed up the tree. Such upgrades can be found via P-Organ. In phase 3, there is the large node "Shorten Fruit Bearing Time", and in the "Item Type" small node there is "Increase max Gold Coin Fruit". The Puppet vendor Polendina will sell the item "Alchemical Booster" (Expanded Wares) which will speed up growth.