Here is the full list of all 52 Crash Bandicoot 4: It's About Time trophies - there are 40 bronze, 9 silver, 2 gold and 1 platinum.
Find the hidden Blue Gem
Pull off a Triple Spin
Defeat Dr. Neo Cortex… again.
Make some noise!
Aim and fire!
Make music in an unlikely place
Beat any boss in N. Verted mode
Find the hidden Green Gem
Find the hidden Red Gem
Scare the cat hiding from a party
Complete a level with Pass N. Play enabled
Play a game of Crate Combo
Play a game of Checkpoint Race
Betrayed! By your worst enemy.
Defeat the Doctors N. Tropy
Grabbed a bushel of Bumpa
Earn any N.Sanely Perfect Relic
Meet Ika-Ika
Beat a level in N. Verted mode
Earn any Time Trial Relic
Defeat Dr. Neo Cortex
Find the hidden Yellow Gem
Meet Kupuna-Wa
Do some target practice
Defeat N. Brio
Complete any Timeline
Shut N. Brio up
Meet 'Akano
Defeat N. Gin and his Weapon of Mass Percussion
Earn any Flashback Relic
Acquire a Flashback Tape
Shut N. Gin up
Complete a level as Coco
Beat a story level without dying
Try on a Skin
Meet Lani-Loli
Crouch or slide into a double jump
Complete a Bonus Path
Spend some time surfin' the waves
Get EVERY trophy
Complete ALL levels in N. Verted mode
Complete all Timelines
Watch the 100% bonus ending
Watch the 106% bonus ending
Earn half of the Clear Gems
Earn ALL of the Clear Gems
Earn ALL N. Verted Gems
Earn ALL of the platinum Flashback Relics
Earn ALL N.Sanely Perfect Relics
Earn ALL of the platinum Time Trial Relics
Defeat an enemy with Triple Spin
Find the source of the Bumpa Berries