Here is the full list of all 92 Idle Gem Quest achievements.
Unlocked the Warrior
Upgraded the Warrior to Level 5
Upgraded the Warrior to Level 10
Upgraded the Warrior to Level 15
Max out the Warrior
Unlock the Bard
Upgraded the Bard to Level 5
Upgraded the Bard to Level 10
Upgraded the Bard to Level 15
Max out the Bard
Unlock the Archer
Upgraded the Archer to Level 4
Upgraded the Archer to Level 8
Upgraded the Archer to Level 12
Max out the Archer
Unlock the Ninja
Upgraded the Ninja to level 4
Upgraded the Ninja to level 7
Upgraded the Ninja to level 10
Max out the Ninja
Unlock the Mage
Upgrade the Mage to level 3
Upgrade the Mage to level 5
Upgrade the Mage to level 7
Max out the Mage
Attack 100 times with the Warrior
Attack 500 times with the Warrior
Attack 1000 times with the Warrior
Attack 5000 times with the Warrior
Attack 10000 times with the Warrior
Play 100 Magic Notes as the Bard
Play 500 Magic Notes as the Bard
Play 1000 Magic Notes as the Bard
Play 5000 Magic Notes as the Bard
Play 10000 Magic Notes as the Bard
Send 100 Arrows at monsters as the Archer
Send 500 Arrows at monsters as the Archer
Send 1000 Arrows at monsters as the Archer
Send 5000 Arrows at monsters as the Archer
Send 10000 Arrows at monsters as the Archer
Slice monsters 100 times as the Ninja
Slice monsters 500 times as the Ninja
Slice monsters 1000 times as the Ninja
Slice monsters 5000 times as the Ninja
Slice monsters 10000 times as the Ninja
Activate 100 Runes as the Mage
Activate 500 Runes as the Mage
Activate 1000 Runes as the Mage
Activate 5000 Runes as the Mage
Activate 10000 Runes as the Mage
Attack 100 times in 10 seconds
Activate 10 musical notes in 1 attack
Land 10 arrows in a row without touching the ground
Perform 100 slices in a row
Cast 10 perfect 'hard' spells in a row
Actively Attack 1000 times while on fire
Actively Attack 250 times while on fire
Actively Attack 100 times while on fire
Obtain a Perfect Emerald gem
Obtain a Blood Red gem
Obtain a Scum Green gem
Obtain a Huge Green gem
Obtain a Priceless Max gem
GPS of over 1 Smoky Purple gem per second
GPS of over 1 Spiky Green gem per second
GPS of over 1 Blood Shard gem per second
GPS of over 1 Pointy Emerald gem per second
GPS of over 1 Lime Dice gem per second
Defeat 4 Monsters
Defeat 8 Monsters
Defeat 12 Monsters
Defeat 16 Monsters
Defeat ALL Monsters
Play for over an hour
Spend 10 Diamonds
Spend 50 Diamonds
Spend 100 Diamonds
Spend 250 Diamonds
Spend 500 Diamonds
Own 1 Artifact Gem
Own 2 Artifact Gem
Own 3 Artifact Gem
Own 4 Artifact Gem
Own 5 Artifact Gem
Play the Monster hunt 1 time
Play the Monster hunt 5 times
Play the Monster hunt 25 times
Play the Monster hunt 50 times
Play the Monster hunt 100 times
???