I don't know squat about Native Americans, but I do about Indigenous Australians, and what little I have read about Native Americans often feels familiar.
Some element of the Indigenous Australian affinity for the environment could possibly be noble savage mythologising and ethnic nationalism. White Australia has always portrayed Indigenous culture as noble savages living in perfect harmony with nature and each other (evident in writing by Cook and Darwin) or as brutal Stone Age rejects (for example, the account of William Dampier, who called them the ugliest and most miserable people on Earth) . The latter view of Indigenous culture was by far the more popular, to the point where states legislated the total assimilation of Indigenous people 'for their own good', and was the norm up until the sixties and seventies, when white Australia underwent a significant cultural shift away from supremacist, eugenic and segregationist belief.
At the same time, Indigenous activists and scholars latched on to the best aspects of European noble savage accounts to counteract the view that the Indigenous were a stone age people who never evolved or created any unique culture or innovations of value. They could hold up a harmonious society that cared for its people and landscape, as told by a few European explorers and colonisers, to contrast themselves with colonial rule, which damaged the landscape, brought in drugs, alcohol, racism and genocide, and in general was nasty and degenerate. Scholars and activists were also usually exposed to and supported by white left-wing ideas, and opposed to the right, so it makes sense that they would emphasise values that would win them left-wing support, like environmentalism. It is also a means of debating sovereignty and self-rule - Indigenous people hold different values, communal values, and thus should not have to live in an Australia where the white majority dictate all norms, laws and values.
That being said, Indigenous Australians were indeed far less destructive and far more willing to work with the environment. For them, it was a life giver that needed to be cared for, especially as Australia can be a fragile place. Ancient soils, flat geography and low rainfall meant you had to be careful with what you had - overuse or spoil it and you starve or thirst. Thus, they had special stories about almost every waterhole and river, every hill and mountain, with each story having educational value, such as "don't use too much water, or it won't refill for next year" or "don't pick all the fruit, or the tree will die". These stories were retold in song, dance and art, connected to famous ancestors or mythical characters, making them easy to remember, adaptable and engaging, meaning some stories lasted tens of thousands of years. Indigenous Australians have stories of when the Ice Age ended and the coastline shrank, and of megafauna 20,000 years extinct, like the giant pythons that are said to have inspired the Rainbow Serpent so central to many cultures across the continent.
This made landmarks and waterways sacred. Every place has a story, a meaning, a use, and an ancestral connection. It was the duty of a nation to care for its land - the death of its people was the death of the land. Care for the land was care for your ancestors. This also meant there was little incentive to invade your neighbours - you can't leave your duty or your ancestors behind, because to do so would destroy your own land. They also received totems at birth, either chosen or connected to family descent, which tell them which animals are especially important to them, and gives them a responsibility to give that creature extra special care. This leads to those people generally being experts in that animal - how to hunt it, heal it, its habitats, etc.
It was different for every nation, but the people who lived in my city, the Noongar, had a six season calendar that dictated where they travel, what they eat, and what would grow. Thus, they never overburdened the landscape, and had enormous diversity in what they eat, leading to them being one of the healthiest people on Earth at the time Europeans began to explore Australia. It's quite odd seeing paintings and photos of first-contact Aboriginals that are as fit as modern athletes, full on six-packs - severe health issues, whether they be addiction or obesity, are the norm today. Some nations lived in permanent or semi-permanent 'villages', and usually these were tied to cleverly designed large-scale fish traps, some thousands of years old. Europeans often couldn't believe that the villages nor the fish traps were built by Australian Aboriginals, leading them to believe that some other culture must have come and gone, maybe killed by the 'savages'.
Another accusation levelled at the Indigenous by Europeans was/is that they were/are lazy, "only picking what nature lets fall to the ground". This is definitely not the case. Bill Gammage in Greatest Estate on Earth talks about how near every explorer who came to Australia commented on how the landscape looked like a perfectly manicured English manorial estate, with lawns and groves artfully place. Grass grew in fertile soil, trees in infertile soil, and soils of the same quality often had completely different plants growing on them - all due to Indigenous landscaping, which Europeans could not fathom. Indigenous Australian lit fires at precisely timed moments in precisely located positions to create favourable conditions for themselves, whether that be grass to grow native animal populations, diversity in edible plants, trees to block animal escape paths, or land clearing for pathways along routes to waterways. Europeans noticed fairly quickly that grassy areas near their farms were soon overgrown bush forest not long after they arrived, and struggled to understand why - it was because they had banned natives from lighting fires and pushed them out of the area or killed them off. Paintings by explorers rarely match the modern landscapes due to overgrowth.
Indigenous farming, mostly of yams, was also widespread, especially in areas that flooded or dried out annually. One tribe went to war with early Sydney because farmers seized their fertile farm land from beside a river. My city was full of swamps and lakes, which were bountiful sources of every kind of resource necessary for life - and in summer, when the water mostly evaporated, the exposed wetland soil was perfect for quick growing 'crops'.
Indigenous people also managed marginal and fragile landscapes far better than Europeans. Landscapes that were grassy and fed decently-sized populations before invasion degraded within one or two generations when farmers moved in to grow wheat or raise cattle, increasing desertification. Even in the central deserts there were tribes planting and farming grasses for flour, which saved the life of many an explorer. One of the most common jobs for Indigenous Australians today is as park rangers, as our national parks are enormous and locals nations have extensive knowledge taught from childhood, and spiritual connections to the land they work on, making them perfect a job somewhat undesirable for urban white Australians. Indigenous people also suffer from extreme unemployment too, so two birds, one stone.
Good resources:
Bill Gammage, Biggest Estate on Earth
Bruce Pascoe, Dark Emu
Sylvia Hallam, Fire and Hearth
Kaartdijin Noongar website.