There are several options to make Lost Ark Gold coins throughout you Lost Ark adventure, but it is likely that you will earn most of them through the game's Island Quests system.
As the name suggests, Island Quests are special quests that you can complete on various islands you can visit. Every island in the game is assigned the capacity of a "cap" that basically determines the amount of Pirate Coins that you can earn on them by performing certain tasks (although there are some rules that aren't followed, which we'll get into in a minute). Here's a breakdown of the many Pirate Coins you can get on each island through quests:
While there are definitely islands in the game that are more lucrative than others, the game's progression system implies it's likely that you'll end up hitting some of islands that are "lesser" islands prior to moving towards the more lucrative ones. In any event, it's absolutely worth the effort to go to each island within the game if trying to maximize your Pirate Coin's potential to earn coins.
They're definitely the most effective method of earning Pirate Coins, but once you've finished them all (which should take you some time) most likely have to depend on Marine events for your Pirate Coin needs.
Marine events are regularly scheduled in-game activities that require you to do a variety of tasks (the most profitable of which involves Ghost Ships). When you complete them (and other actions that you can do in the game) may reward players with unique Voyage Coins that can be exchanged in exchange for Pirate Coins at the following exchange rates. In other words, you typically can expect to get a few more Pirate Coins by completing certain nautical tasks. But, if you're looking in making as many coins as possible as quickly as possible then you'll need to be able to make the most of your time best place to buy Lost Ark Gold.