Do green roses exist

In the wild, no they do not. However, they can be grown fairly easily. If memory serves, you make a container (able to hold water) that has thin, watertight dividers. Then you carefully split the stem of the rose (near the bottom, only split a few inches, otherwise you'll probably damage it) so that it has the same number of stems as the container has sections. Then you fill each section with water dyed a different color, and put the rose in the container so that each piece of the stem is in a separate section. This must be done very carefully so as to avoid breaking off pieces of the stem.

Why it works: The stem, of course, pulls in the water to keep the plant hydrated. Much of this water goes to the petals, which in turn gradually take on the color of the water.

I can't guarantee this method will work the first time, as you may have to practice the delicate art of dividing the stem and then placing it.