The expression green around the gills means someone is ill from motion sickness, the flu, or perhaps from too much alcohol.
I'm not sure when Go Green first became a battle cry for environmental movements.
My guess would be that it was sometime in the early 1970s that Go Green started being adopted as a slogan for anything from ocean conservation groups to recycling efforts.
Sometimes go green becomes ironic because it's used by commercial enterprises that aren't particularly good for the environment. They want you to think their products are better for the environment than their competitors products and begin their ad with GO GREEN.
Language and expressions can be confusing but tone of voice helps. If I say I'm going green and sound like I'm about to pass out that probably means I'm green around the gills. If I say I'm going green with a happy tone of voice, you can assume it has to do with an environmental issue I'm trying to help with my genuine efforts.
Gee whiz on with the games . . .