Gems can be found in all game modes. Each game style has its own gem:

GemLocationGame styleProbability
Topaz TopazStandard mode, standard difficultyEfficiency
0.001 x CEff x Difficulty
Ruby RubySpeed, FL
0.001 x 3BV/s x Difficulty
Sapphire SapphireSpeed, NF
0.001 x 3BV/s x Difficulty
Amethyst AmethystWin streak
0.001 x CWS x Difficulty
Onyx OnyxStandard mode, high difficultyHigh difficulty
0.001 x Difficulty
Aquamarine AquamarineNo guessing mode
0.001 x Difficulty
The probability is doubled for games with win streak.
Emerald EmeraldSeason rewards
Garnet GarnetQuests
Jade JadeArena
Diamond DiamondExperience
1 diamond for every 10K exp

CEff Efficiency coefficient (Efficiency/100+0.05)10
CWS Win streak coefficient Depends on win streak value and board winrate.