Run the game in the background. Every 24 minutes (of real-life time), tab into the game and do the following: 1. Talk to the fishing NPC (Johnny) and find out what fish he wants. 2. Go to the chest labeled with the current fish and check its content. 3. If there's fish in the chest, take one and turn it in to Johnny. 4. If there is no fish in the chest, right-click the accelerator marker on the railroad right below the chest. The railroad will take you to the place where you can fish up what Johnny is asking for.
At every fishing spot, there's a chest labeled "Fresh Catch". You can only catch the quest fish if you don't already have one in your inventory. When you catch the quest fish, place it in the chest and continue fishing. After you've fished up 10-15 of the quest fish, move them all from the chest to your inventory and use your mirror to get back to the village. Turn one of the caught fish to Johnny, place the rest in the appropriately labeled chest.