I can't find anything either which would support his contention. I'll keep looking. I wonder why he would claim such a thing. However, on Wikipedia there is something that comes close. "After the Roman conquest of Greece, an unofficial diglossy of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire."
This rather overstates the case, in my opinion. I think it's true of the patrician class, since Greek was a basic part of Roman formal education. In fact, many tutors were Greek slaves and Romans who wished to pursue higher studies in philosophy regularly went to Greece to do so. But patricians were only a minority of the population, and most Romans were educated by their parents within the family, if at all.
So I can see the officers of the Roman army all knowing Greek (in something like the way their British counterparts two millennia later all knew Latin). But I wouldn't expect it among the rank-and-file unless they'd been recruited specifically from those parts of the Roman Empire where Greek was the lingua franca.