You have to unlock time attack mode for this which means completing the game once.
The first available special arrow is the whistling arrow, which is rewarded for killing 2 colossi in Time Attack mode. That mode can only be accessed on Normal or Hard difficulties.
To fight one in time attack , press R2 to pry in front of their statue in the temple. I recommend the 1st two (ones nearest the front of the temple) on normal. That said you'll have to beat them all for the other trophies anyway. Once done go pick up your reward from the pool at the back of the temple. Press the touch pad then triangle to make sure the new arrows are active and back out and press left on the d-pad to select them. Best spots for lizards are the shrines.
There was a delay in this popping on screen for me , maybe 15-20 seconds after shooting the lizard.