Here is the full list of all 23 Cat Quest (JP) trophies - there are 7 bronze, 7 silver, 8 gold and 1 platinum.
MEOW!
Completed the Main Quest
Reached level 99
Obtained 66 armor or weapons
Cleared all 52 dungeons
Completed all 62 side quests
Defeated the Arcane Dragon
Defeated the Ice Dragon
Obtained all 7 skills
Located all monuments
Reached level 50
Cleared 25 dungeons
Obtained 4 skills
Obtained 30 armor or weapons
Defeated the Flame Dragon
Located half the monuments
Completed 30 side quests
Reached level 10
Cleared 5 dungeons
Completed 6 side quests
Obtained 2 skills
Obtained 3 armor or weapons
Located one monument