I just completed all of the quest related badges in Pirate101, but only the Book 12 badge (Hero of Subata) gave me issues. If you too are having issues, check out the below NPCs as they might also be your problem like they were for me.
Brother Hei Yu (Wai No Shrine):
His location in the shrine can sometimes be missed off to the side.
Yagi Sonoda (Temple of the South Wind):
Her location on the far right of the temple is nowhere near any main quest NPCs and her quest is usually missed by those who pick up every quest.