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Green Chests are magically locked treasure chests that Swaine can open using his lockgun. To do that, you need to locate a spot where Swaine can fire the weapon while facing the front of the green chest. That clue should help you narrow down the spots you need to search for a good shooting spot.
Green chests are found in dungeons and other interior areas (like towns, caves, etc.). Horace's Chart Chests spell (complete Horace's Riddles to earn it in Hamelin) will let you find all manner of chests in interior areas.
See Also
Ni no Kuni - TV Tropes