I agree with funkyanarchy's comments, but I'll bite for the actual elaboration of the question, but I need to clarify a few things first:
1. When I write about a "green anarchist society," I am explicitly referring to anti-civ tendencies (but not necessarily exclusively anarcho-primitivist) which assumes an ongoing project of de-industrialization (ongoing because I don't believe there will be one big massive collapse where suddenly it is all gone and we are feral psychic monkeys).
2. I take issue with the singularity of "green anarchist society" because I think the idea of a green anarchist potentiality implies multiple societies, or maybe more accurately cultures and communities (both of these terms are also not without baggage, but that is something for elsewhere).
3. There are probably other assumptions I am making that are not totally clear to me right now, and I will edit this to reflect them as they come into focus.
In a green anarchist society, protecting the environment would likely not be a problem for multiple potential reasons. Essentially I think it boils down to questions of scale, capability, and priorities.
One of the primary ones is that environmental destruction on the scale that we currently experience is unlikely to continue with he removal of industrial production and the requisite resource extraction. Depending on how Shit Goes Down there might be a temporary spike in people scrambling to gather/horde resources, but as the man-made systems that sustain rampant human overpopulation fell away (and the population decreases with it*) that would taper off, hopefully before things were so far out of balance that there was a massive inter-species die-off (though at the point we are at now, it isn't clear that we would avoid that even without a sudden and temporary increase in such). At the point where things leveled out, humans would lack both the technology and the numbers to continue to do the worst of what we are doing right now.
Also, we if we are talking about a green anarchist future, then I assume we are talking about very localized groups living in relation to their land-base in ways that are radically different than current reality of those of us who are civilized (even if we are totally down with rewilding in our personal lives, we are still inhabitants and products/victims** of civilization). I expect this would greatly inform how we choose to act on/with the rest of our environment in said future. Many AP folks would probably suggest that the very dichotomy of humans/environment would not exist anymore, I am less sure, but think it is worth thinking about.
The reemergence of mass culture, capitalism, industrialization, etcetra is always a possibility. I think this is one of the places where recognizing that most anti-civ folks aren't longing for a return to a pristine and innocent state (some sort of pre-fall Garden of Eden situation), so, one would hope, memory of the dangers and pitfalls of such a world would be remembered and combated by whatever means necessary. Still, the problem remains: it is possible that some group would start to horde resources, develop surplus, expand population, and encroach on/destroy/enslave other groups, eventually gaining near or total domination. Personally, I think the odds are better that there won't be a mass consumer system (and the resulting rampant environmental destruction) if we remove the one we have now than if we carry on with what we have now or act in half-measures.
* if you were thinking about accusing me of advocating genocide here, I know... consequences...
** I chose the word victims because I don't think we have come out the other side to the extent we can call ourselves survivors.