There are many possibilities to get Lost Ark Gold through an Lost Ark adventure, but the likelihood is that you'll earn most of them through the game's Island Quests feature.

As the name suggests, Island Quests are specific quests that you'll need to complete on various islands that you are able to visit. Every island in the game is assigned an "cap" which is basically how many Pirate Coins that you will earn from completing specific quests (although there are rules that aren't followed, which we'll discuss in a minute). Here's a breakdown on how many Pirate Coins you can earn on each of the islands through quests:

While there are obviously some islands in the game that are more lucrative than others, the game's progression system is that you'll likely be able to hit some of more "lesser" islands initially before moving to the more lucrative ones. However, it's definitely worth hitting every island in the game if you're hoping to maximize your Pirate Coin's earning potential.

These quests are certainly the most effective method to earn Pirate Coins, but once you've completed them all (which is likely to take an extended time), you're probably going to have to rely on Marine events to fulfill your Pirate Coin needs.

Marine events are often scheduled in-game events which require you to do a variety of tasks (the most profitable ones are Ghost Ships). Completing them (and other activities that can be found throughout the game) might reward you with special Voyage Coins that can be traded with Pirate Coins at the following exchange rates. In short, you are likely to earn a few extra cheap Lost Ark Gold by completing certain nautical tasks. If you're interested 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.