Quite a few sources say that Aristotle was the progenitor of the term, after his use of energeia (a Latin transliteration of ἐνέργεια). However, one source notes that Heraclitus used a similar word, en-ergon, years prior. Heraclitus wrote:
"En-ergon is the father of everything, king of all things and, out of it, all forms of contrast originate. Since ‘en-ergon’ is common to everything, it is vital for life itself.”
This was part of his aforementioned flux theory. However, like Aristotle's ideas, his concepts bear no relation to our modern usage of 'energy', and I doubt the two are related. It is worth noting, though, that Thomas Young's coinage of 'energy' in the early 19th century is thought to have been influenced by Aristotle's usage of energeia. This would mean that the two, while not the same idea, may be thought of as true cognates.
I’m going to compare some ancient ideas and explain why they aren’t really precursors to our modern idea of ‘energy’. I suppose this is a "Too long; didn't read" summary; a lot of the other things are extraneous. Note: All information is taken from sources given later in the answer., unless otherwise indicated.
- Qi: This is occasionally miswritten as 'Qi energy', which is redundant. Qi is often translated as 'natural energy' or 'energy flow'; adding 'energy' to the end would be incorrect. Qi was the ancient Chinese idea of an all-permeating substance in every living creature. It was responsible for life living. Obviously, this is not 'energy' in the modern sense because it is not conserved (dead things have no Qi).
- En-ergon and the flux theory: Heraclitus’ brainchild was the flux theory, which states (explained better here, but sometimes the page goes down and is inaccessible) that there is some substance that permeates all objects. The reason this isn’t ‘energy’ is that it is uniformly distributed – that is, an object sitting on the floor won’t necessarily have a different energy than an object rolling on a table.
- Energeia/ἐνέργεια/potentiality and actuality: This was Aristotle's idea. It comes nearer to energy than Heraclitus' ideas, but closer examination reveals major differences. For example, a tree always has potential - to become a wooden bowl! Likewise, a person has potential to be married. These are really just possibilities, not 'energy' found in objects. Aristotle's ideas are not much like modern scientific ideas of energy.
- Vitalism: This idea, apparently coming first from Aristotle and surviving for about two millennia, was similar to Qi - actually it was nearly identical - in that it was the idea of a 'life force'. According to Wikipedia, it gained the support of eminent scientists such as Jöns Jacob Berzelius, one of the foremost figures in 19th century chemistry.
- Vis viva: This really isn't part of the list, but it's a true precursor to energy, and worthy of discussion. The term vis viva was coined by Leibniz in the 17th century, to describe the "living energy" (vis viva is Latin) of a moving object. This truly was a precursor to kinetic energy, and I'm actually a bit surprised that our 'energy' isn't actually a lingual descendant of vis viva.
After a week or so of research, I found just about nothing that directly confirms or denies the possibility presented in the question. Zip. Zero. Nada. I've gotten the only null result I've ever gotten in 4 months when researching an answer. It's frustrating, but it's a bit helpful, because it taught me that there are some things that you may never know, no matter how hard you try to find them.
Back to the null result. The question asked whether or not any non-European, ancient cultures came up with something related to our modern concept of energy. The answer, as far as I can tell, is that nothing of the sort existed before a few centuries ago.
If you'll pardon this little detour, I'll talk about Europe first. As fdb hit upon, Aristotle is often cited as the creator of a concept similar to energy. I discussed it in this answer. The problem is (which I should have expanded upon), actuality and potentiality were philosophical ideas. Sure, loose interpretations could make them seem similar to the "energy" of today, but they were quite different from our modern concept. By Aristotle's logic, a tree resting at what we would call a reference point (i.e. where potential energy is 0) would still have potential - the potential to become a wooden bowl!
Next, we go to Gottfried Leibniz and the idea of vis viva (behind a paywall and/or registration, but the free first page should give you an idea of what I'm talking about). He argued that a moving object was imbued with some sort of quantity. This could be interpreted as either kinetic energy or momentum, and in fact there was a dispute over this. We might also say that Descartes was also involved. Nevertheless, 350-ish years ago, there was some idea (in Europe) that a quantity in moving object was conserved.
Going to Wikipedia (a short and mostly irrelevant detour, I assure you), we find that the term "energy" (and then "potential energy" and "kinetic energy") was not coined until the early 19th century, by a few separate scientists working on the idea. At this point, we may consider the concept of energy (in its modern incarnation) to have been born. If you're fine with Wikipedia, you can read more here.
Try a few searches (duckduckgo is my engine of choice) on the internet. I recommend the following:
- History of energy
- History of energy concept
- Energy concept
- Energy ancient [culture]
- Discovery of energy
- Energy ancient science concept
. . . and many more. All turn up results either about energy generation or (once or twice) crank sites about weird life-force ideas that are decidedly kooky. I was not able, in fact, to get a single mention of ancient cultures when it came to the concept of energy. This is pretty bad, because a result of "No, there was no idea of energy in ancient cultures" would be definitive. Here, there is nothing.
The one exception I found was from this site, which gave the completely following (no sources were cited):
Ancient cultures associated fire with all forms of energy, and the Greek philosopher Heraclitus (~500 BCE) explained everything in the universe in terms of energy. In the centuries that followed, many scientists proposed theories to explain the concept of energy, but few shed any true light on the subject.
Unfortunately, the "ancient cultures" are nowhere to be found. Go ahead a couple hyperlinks to this page. Heraclitus' flux theory is mildly interesting but not too relevant. Further down, we see hints of something along the lines of conservation of mass, but it goes nowhere. The knowledge that Aristotle disagreed with him leads me to believe that there is no hope of a connection between the two.
Just for fun, I went to the Wikipedia page on Qi. The on thing that caught my eye was this sentence:
Notions in the West of energeia, élan vital, or "vitalism" are purported to be similar.
Going to the pages listed reveal nothing of any importance (and none of the ideas are anything like our modern idea of energy), but the fact that Aristotle is listed nearby further convinces me that his work is not really scientific.
One last stop. Many searches brought me here, where the author (see the PDF link on the upper left; I've had a hard time copying it here) claims that Joule and Mayer were the originators of the idea of energy. It seems like they certainly made some empirical investigations, but the ideas of Descartes and company seem to predate these men.
I'll end with Bertrand Russell. For those who don't know, Russell's Teapot is the idea that the burden of proof is on the claimant. I could say that there is a teapot floating between the Earth and Mars; it's nearly impossible to disprove unless you search every corner of the space between the solar system and so I can say that it exists. Our situation is similar to that, in that the idea that ancient cultures had a concept similar to energy is hard to disprove. One the one hand, I have found nothing; on the other, there are still chances that I am completely wrong. I'm going to let the community judge this answer and its validity, but at the moment, my answer to the question is "No."
By the way, I'd love some positive and/or negative feedback here. This was an odd experiment for me, and I'm not sure how well it worked.