Head to the enemy base found north of the snowy mountain region. There you will find the depresso fishing rod recipe which is the best rod in the game. Craft that with High Quality Pal Oil x20,
Paldium Fragment x140,
Pal Metal Ingot x30,
Carbon Fiber x20,
Coralum Ingot x5.
Then get Whalaska and Whalaska Ignis for easier fishing and Jelliette for better fishing drops. Then find very purple fishing spots and you will have a decent chance to find a lunker.