Not that I know of, although the sort of sentiment behind it was common. In Greece, at least classical Athens, there was almost a cult of the soldier, with, for example, Aristotle claiming that the soldier demonstrates the apogee of virtuous courage, and the playwright Aeschylus' epitaph famously only said that "I fought at Marathon" (an equivalent of this would be if, say, after Mel Brooks died his memorial only mentioned his service in WWII). There was a certain landscape psychology to this as well--in the ancient Mediterranean cemeteries were typically immediately outside the wall of city limit, and generally clustered on the main roads leading out. Among these were many military memorials, and so a somewhat clumsy comparison is if a significant portion of the population went by Arlington Cemetery on a regular basis.

Rome is a bit more complicated, because when we start to get good information about it, it is not a city state, but rather an empire. One major difference arising from this is simply geographic: a typical person in the Roman empire would be much less likely than a typical person in Athens to come into contact with a veteran. Regional military recruitment was disproportionate, so there were some areas such as Batavia where military recruitment impacted virtually everyone's lives, but these were fairly unusual (and led to a full scale revolt in 69 CE). In the United States one of the factors in how we perceive the military is a sense that we are all one big community, and soldiers are members of that community and their experiences are part of the greater communal experiences--hence they are "our boys". The Roman empire by contrast was not seen as a single big community, it was seen as the area subject to the power of Rome (Rome here being sort of the city itself, but also sort of not, the orator Aelius Aristides said that the emperor made all worthy men Romans). So someone in Syria reading the Damascus Tribune^1 in 9 CE might think that the Germans destroying three legions might think it is pretty scary and pretty bad, but would not necessarily think that "our boys were chopped down in the woods".

On a practical level, this also means that the subjective experience of encountering the Roman military was different than it is for us today. One of the best examples of this comes from a part of Apuleius' Golden Ass in which the narrator meets a soldier (the narrator is a donkey, long story). The soldier is portrayed as part of the world in which people inhabit, but not part of the community: they are Romani and not Graeci, they speak Latin, they live separate and there is a wide social gulf between them. This should not be taken to mean that Thessalians should be thought of as an "occupied" population and certainly the text should not be read as anti-imperial, but the soldiers are outsiders. And in reality, military colonies tended to be pretty insular places, research in African colonies shows that intermarriage between soldiers and locals was very rare.

So in general while the Athenians did indeed have a relation to the military not unlike ours today, but Rome was a very different sort of society.

^1 Before I get in trouble for this, there was no newspaper in ancient Damascus, and if there were it would probably not be called Damascus Tribune. The closest thing would probably be the Acta Diurna in Rome and similar things, and those would have been more like public notices.