2 [in singular] The nature or essence of a person.
‘sometimes one aspect of our being has been developed at the expense of the others’
The phrase "the quiet [being] of the inhabitants of nature" strongly elicits the definition above. That is, it tends to read as "the quiet essence of ...".
Since the comments suggest that you are distinguishing between existence and living, and wish to express existence in your sentence, the sense you're trying to elicit is slightly different. You're referencing the continuous tense of be (coloured in this case as mere existence):
‘I think, therefore I am’
You also ask:
Can I write "being" as "be-ing"? Is that even an acceptable way to write the word?
Consider this from Grammar Girl, a respected writer on grammar:
Sometimes, the placement of a hyphen changes the meaning of your sentence. Let’s say you want a “hot-water bottle.” With a hyphen between “hot” and “water” you clearly want a water bottle for holding hot water because “hot” and “water” are joined by a hyphen. Without the hyphen between “hot” and “water, you might want a water bottle that is hot. See how the presence or absence of a hyphen could change the meaning?
The more likelihood there is for confusion, the more you need a hyphen.
YourDictionary.com says something similar:
The fact is, there’s really no set of hyphen rules on which every person can agree.
Hyphens can be useful to avoid confusion.
From the above, I'd argue that you should use the hyphenated form be-ing in your context because it elicits the 'be' sense more strongly than the 'being' sense (with respect to the two definitions cited above). It's a legitimate use of the hyphen (avoiding confusion) because the 'being' sense would otherwise dominate.