Simply load up Seattle Day 1 - The Birthday Gift
Once you enter the museum, look to your right and grab the hat (fedora) hanging on a rack.
Go down the hallway, look to your left and put the hat on the Stegosaurus. Then, look to your right and put the hat on the Triceratops. Finally, put the hat on Joel and after some cute dialogue, the trophy should pop.
Hope this was helpful and happy trophy hunting!