As an ecosocialist, of course I think that this is the case, but not everyone who is a green or a socialist agrees with this thinking. It does depend on what you mean by green and socialist in the first place though. Many greens are essentially liberals or green social democrats, and many people who call themselves socialists are in fact social democrats, or as the phrase popular in the US these days has it, democratic socialists.
Conventional socialists, all too often see ecological politics as an add on issue, at best, which will somehow cease to be a problem when we move to a traditional socialist system. Socialist governments that we have seen around the world (USSR, China etc), have often had an even more dismal environmental record than capitalist countries.
Ecosocialists think that both are wrong.
Ecosocialists see the problems of ecological justice and social justice, as profoundly linked in our economic system, capitalism, and no amount of tinkering with capitalism will ultimately resolve the ecological and social problems caused by this unfair and destructive system.