According to Carl B. Boyer, "The history of the calculus and its conceptual development", Dover Publications 1959, page 98,
The improved notation led also to methods which were so much more facile in application than the cumbrous geometrical procedures of Archimedes, of which they were modifications, that these methods were eventually recognized as forming a new analysis—the calculus. The period during which this transformation took place may be considered as the century preceding the work of Newton and Leibniz.
(The question is complicated by the fact that most mathematicians were writing in Latin.) If you are asking when the word "calculus" was used to refer to the differential and/or integral calculus, that must surely post-date the development of those subjects. For example, Richard Suiseth was known as Calculator in the 14th century, and experts on what we now call arithmetic were called "reckoners" in the middle ages. The word "reckon" is really the German word "Rechen", which means "calculate". So "Calculator" is just Latin for "Rechner". So the word "calculus" in one form or another was around for a while in regard to arithmetic. The question is somewhat confused by the fact that what we call "mathematics" was actually called "geometry" until only a few centuries ago, and mathematicians were called "geometers", even into the 19th century. The term "mathematics" in Greek refers to knowledge in general.
Perhaps the better question to ask is when were the terms "differential calculus" and "integral calculus" first used. The guy who taught Newton his basic calculus (before Newton developed it further) was Isaac Barrow, who wrote a book "Geometrical Lectures", showing that this subject was still considered to be part of geometry at that time. According to Boyer, page 190, Newton used the word "analysis" for his first publication on calculus.
The first notice of his calculus was given, however, in 1669, in De analysi per aequationes numero terminorum infinitas.
And integration was originally called "quadrature", as in Newton's De quadratura. (Boyer, page 205.) It seems to me that it was Leibniz who first promoted the term "calculus" in the modern sense. He wrote a book Historia et origo calculi differentialis a year or two before his death. (Boyer, page 215.) In 1712, the Royal Society committee reported that
The differential method is one and the same with the method of fluxions, excepting the name and mode of notation [...]
(Boyer, page 221.) This shows that Newton was not even using the term "differential calculus". (The term "differential method" referred to Leibniz's method.)
According to Boyer, page 67, the word "integral" originated with Leibniz also, but on the suggestion of the Bernoulli brothers. So it seems to me that the term "calculus", with that name, developed only during the late 17th century, more in Europe than in Britain, where the term was resisted for some time.
PS. If you just want to know when the word "calculus" was first used for calculating (as opposed to the meaning of small stones or pebbles), my Latin dictionary (John T. White, 1902, 1923) gives quotes from Cicero and Livius where the word was used in the sense of "reckoning, computing, calculating". That puts it back to 2000 years ago, plus or minus about 30 years.
PPS. Perhaps I could also mention that what was called "calculus" in ancient Roman times and up until the Renaissance in Europe was not considered to be part of the same subject as geometry, which was taught at the universities and the precursors of universities. There was a sharp divide between the advanced mathematics of Euclid, Apollonious and Archimedes and the market-place arithmetic which is well-documented since the beginning of writing 5500 years ago. Everyone in these 5.5 millennia would have learned the basic marketplace arithmetic as part of growing up. It was not a study as such. It did not need a university education. Nor was it generally learned from books. So to call that marketplace "calculus" or "reckoning" part of mathematics is not really correct. If you ask a mathematician right now to add up a supermarket invoice with 30 items, they might have some difficulties, but we would not think that this indicates a lack of mathematical ability. That's just the "reckoning" of the marketplace, and carpentry, book-keeping, and dozens of other trades. So in my opinion, "calculus" was not used for a really mathematical subject until the 16th or 17th century. Until then, it was only applied to marketplace addition, subtraction, multiplication and division. Calling that mathematics would be like calling a road-sign literature.