This can only be done on the final hunt, which requires 30,000 points to unlock. Equip all licenses, a sniper rifle and all equipment to get Geared Up at the same time.
The 6 dinosaurs are Stegosaurus, Ankylosaurus, Ceratosaurus, Parasaurolophus, Triceratops and T-Rex. Aim for the heart to conserve bullets, use your Decoy to find missing ones and watch out for rogue Ceratosauruses! Note that you dont need to "collect" them by sending them to your trophy case, just going near them to "confirm" them (and get your gem reward) is enough. The trophy will pop instantly upon confirming the 6th dinosaur.