They would leave the same artifacts humans would have, up to the point where they decided to only use biodegradable everything.
I don't like this answer, but it fits.
This question begs the question, I suppose, of how they got to such a level of technology. If they were always eco-mentalists, why? When you're struggling to survive against nature, privation, tyranny, genocide, etc, are you REALLY going to give a fig about your carbon footprint?
So, understanding that some questions get asked without fully understanding the question behind the question, if your question is "how do I have an untraceable race of greenies," you have to explain not just how they advanced past the Stone Age without leaving lasting traces, but WHY.
Sub-questions: "How can modern humans find the traces of existence of this civilization? What is the most possible and realistic way to prove, that this civilization existed?"
The answer to these sub-questions is essentially the core answer above. Modern humans could find the traces of existence of this civilization in the same way modern humans find traces of the existence of other civilizations: By the parts that didn't degrade into "noise." The most possible and realistic way to prove that this civilization existed is the same way we prove that other civilizations existed.
Keep in mind, religious proscriptions are about things people are liable to do. Few and far between are the explicit proscriptions against eating feces, because that avoidance is built into (essentially) everyone. Any proscription against pollution which is intended to include all artifacts (including trails, bones of the dead, the indentations which homes leave in the terrain versus the dirt that would build up around them etc) from before this proscription would be fighting against human nature. It would also require gargantuan, literally incredible effort. There would need to be a god-agency (which you have) not just making declarations and punishing, but also deeply involved depending on just how wide-ranging this proscription is. I have assumed complete and total erasure, since the question and sub-questions presume it will be extraordinarily difficult.
For example, even if we ignore everything from before this culture gets all nano-techy, the fact that their machinery crumbles into dust does not mean it is untraceable. There will be, after all, a rather conspicuous arrangement of dust. That dust will be made up of whatever those nano-bots were--so if the manufacturing process reduces everything to carbon in order to make nanobots out of carbon nanotubes (or just "alters the atomic structure into MacGuffinite, which is ideal for nanobots"), you'll have a pile of carbon/MacGuffinite. Wind will not be sufficient everywhere to disperse this. If you've ever seen intelligence analysis of aerial recon, you'll see that very vague shapes can be determined to be specific objects by their arrangement, positioning, and other artifacts around them. Archaeology cranks this up to 11.
OK, you can assume perfect cleanup protocols for the last nanobots... well, that's up to you, but then you're stuck with your core question of what I'm assuming is your plot hook. To this I re-state the first sentence: Leave some evidence of former eras for your modern humans to find--especially since removing that would require insane effort even for nanotechnology (aka scifi magic, no offence). Even removing evidence of former eras should leave evidence that something has been disturbed, which can lead to reviewing the arrangement of that something, questioning locals (if not on the planet in question, then perhaps others in "this sector"), analyzing flora/fauna patterns, etc.
Not sure if there's an archaeology SE, but that's the road you seem to be headed towards if you want excruciating details. Read an article every so often from http://www.archaeology.org/ and you may find some great ideas.