There was a conference looking at this kind of question which I went to in March: Greening the Gods. That should give you some sources to follow up.

I have the notes I took, although I'm a theologian and environmentalist with very little classical training, so quite a lot, particularly on Platonism, went over my head.

In broad summary, there are quite a range of views both among different Roman/Greek thinkers, and in opinions about individual ones as to whether humans are an integral part of nature, or more separate. Most of what we have we have to infer from comments on worldview derived from works on other topics, including plays. However, if you had to identify a "middle way" amongst all these thoughts, it would be reasonably holistic - we are part of and derived from "nature" (probably "nature" as distinct from humans wouldn't have been a term they understood), and care for nature would be a positive virtue, although probably more out of a general idea of "restraint" than a fear that nature might be damaged. There was also an overlap in some cases at least between the natural world and the "sacred" - so preservation may have been carried out out of respect for the divine instead of, or as well of, for nature.

That's the best I can do with the time I have available now, anyway. I can write up more about specifics if you're interested.

In terms of "conservation movements" in anything like the modern sense of recognising environmental damage and wanting to do something about it, I'm fairly sure the answer is no: I'm not aware of anything before the mid 17th century when the Royal Navy began to worry about tree shortages and started managing forests. Of course, many cultures have had traditional practices for managing agriculture or forestry to help preserve them, if you want to count those, and I think there's a reference in one of the papers to how the sacred olive groves somewhere in Greece were managed - although as previously noted, whether this was done for devotional reasons or care for nature or both is hard to discern.

Virgil's Georgics arguably make some ecological claims. This is an idea that's fleshed out in literature about Virgil's re-interpretation through early modern English writers, like Milton and Marvell.

See: Poetry and Ecology in the Age of Milton and Marvell, DK McColley

I have a vague recollection of the Roman conception of ecology and biology being very briefly discussed in Nature's Economy: A History of Ecological Ideas, which I had to read part of in high school AP Bio (don't ask). I don't think the Romans did had any ideological or intellectual framework that closely resembles our modern conception of "ecology", although they had some ideas regarding sustainable land use that might fall under the umbrella of human ecology these days, depending on how such a subject is defined.