Already before the ancient Romans, Aristotle had made a classification of the parts of speech based on their meaning as exposed by the kind of questions to which they can provide answers (the so-called "Categories" of Aristotle). For some of his categories he has coined new names, including the name that has been later translated into Latin as "quantity", and Aristotle has divided the quantities into discrete quantities (providing answers to "How many?") and continuous quantities (providing answers to "How much?"). According to Aristotle's definitions, "zero" was certainly a quantity, together with the other numbers and with other words that can express quantities.
I do not think that there is any preserved text from the Classical period of Ancient Greece in which it is stated whether they thought that zero belongs to what they called numbers (arithmoi), or not. In any case whether they had chosen to call zero using the term "number" or not is less important than the fact that they understood perfectly the more general concept of "quantity", which included various subclasses, like numbers (what we call integer numbers) and measures (what we call real numbers), and to which zero also belonged.
About the ancient Romans, the most frequent way of saying in Latin "some", i.e. an unknown number, but at least 1, was "not zero". Therefore regardless whether they were using the word "number" for zero or not, they had a clear concept of a set consisting of 0, 1, 2, 3 and all the other cardinal numbers, so they identified the subset with 1, 2, 3 ... as the complement of the subset containing zero.
The ancient people were much less ignorant than some modern people believe.