Replacing diode in laser
Because the document you linked is for a laser light show device, I'll assume you want to use this for a laser light show.

In that case, you probably do not actually want the green laser to have equal power to the red laser. This is because of the luminosity curve, which shows how efficiently the human eye responds to different wavelengths of light:


(Image source: Wikipedia)

This shows the human eye is near peak efficiency for green light (around 550 nm) and only about 20% efficiency for red light (around 650 nm). So for two laser beams to appear equally bright for human viewers, the red beam should actually have about 5x more power.

That said, if you just want the green beam to be brighter for some reason, you may be able to replace the diode as you suggest. You'd need the be sure the supply is able to provide enough current for the new diode, you'd want the new diode to be in the same size package as the old one, you'd want to be sure to adequately heat-sink the new device when you install it, and you'd want to be sure you can align the output beam of the new diode to the optical projection system correctly. All of these things require knowledge that might only be available in the maintenance manual of the projection system, if they're published at all.