Any color hair can take on a greenish tinge after a lot of exposure to swimming pool water, it's just that the greenish tinge is easier to see on light-colored hair than on dark-colored hair.

Chlorine is not green, and it's actually not the chlorine that causes the color, although the chlorine may help indirectly. Here's what I mean by that:
The green color is caused by certain heavy metals, mostly copper, that lodge in cracks in the scaly outer covering of your hair shaft. This outer covering is called the cuticle, and it normally protects your hair. Swimming a lot can damage your cuticle, allowing the copper and other metals to get it and stick there. Once there, they oxidize (kind of like rusting), and oxidized copper is green. (Incidentally, that's why the Statue of Liberty is green, and why old pennies are sometimes green--the copper in them has oxidized.) Chlorine's role in this is that it helps damage your hair's cuticle. So, although the green you see is not chlorine, the chlorine probably helped the copper get in and stick there. The copper then oxidizes, and voila! Green hair.

Many people have experienced blonde hair going green after prolonged exposure to chlorine in swimming pools. Sometimes darker hair can also develop a green tint to it. The problem is due to high concentrations of copper compounds dissolved in the pool water. This can chemically interact with chlorine and the resulting chemical compound readily binds to the hair. It has also been reported that high levels of copper in tap water can also turn hair green.

Chlorine in swimming pool water may affect the general appearance of the hair. After a swim in chlorinated water most people's hair looks very dull and dry. This is usually due to removal of oils that coat the hair to give it a shiny look and certainly chlorine is a very powerful remover of hair lipids.

So the short answer is, no, it is not because chlorine is green. In fact, it is because the copper compounds, which are also in pools, are greenish. When that reacts with the chlorine it can bind to hair causing it to look green. You can see this for yourself... take a clean copper penny and put it in a little chlorine bleach overnight and see what happens! The penny needs to be shiny clean, not dark looking. Also, please be sure to do this with an adult because chlorine bleach is a dangerous chemical.