If they were Armenians living in Armenia, then normally no, they would just think of themselves as Armenians. "Armenia" could be defined in lots of different ways but generally it covered a much bigger area than the current country. But however big it was, it was always sort of a far off borderland. Early on it was an independent state and just a client of Rome. Later it was integrated into the Empire and divided into the various Armenian "themes" or provinces. Sometimes Armenia was part of the empire, sometimes it wasn't, sometimes parts of it were and other parts weren't. After the Turks conquered Anatolia in the 11th century, there was little or no geographic connection to Armenia, and it was either part of the different Turkic states, or an independent kingdom.

The Romans/Byzantines thought Armenia was kind of a backwater and anyone from way out there wasn't really Roman. But there were lots of Armenian settlements elsewhere in the Empire, and Armenians who lived there, who integrated into Greek/Roman society, and especially the ones who part of the army and the aristocracy, did consider themselves Romans, just like anyone else. They still would have been considered culturally different. They spoke a different language, and their version of Christianity was not quite the same as Greek Orthodoxy. They stood out a bit!

Over the generations, an Armenian family could fully integrate into Roman/Greek society, and they would think of themselves as Roman and everyone else would think so too. Some of them even became emperors - Leo V, John Tzimiskes, Basil I, etc. Basil was from an Armenian family that had settled in Greece, for an example of Armenians living on the opposite side of the Empire.

This issue always makes me think of this quote from Warren Treadgold:

“Being a Roman…was mostly a matter of culture and religion. It was not primarily a matter of speaking Greek or of living within Byzantine territory, and it had nothing to do with race. Byzantium always had Slavic, Vlach, Albanian, or other subjects who spoke little or no Greek, but were Byzantines nonetheless. If an Armenian, Italian, or Turk settled in the empire, joined the Byzantine church, and adapted to native ways, he passed for a Byzantine.”

(A History of the Byzantine State and Society, 1997, pg. 804-805)

The most informative study that I know of is the long article (or very short book) The Armenians in the Byzantine Empire by Peter Charanis (1963), although it’s a bit dated now. He concludes (pg. 56) that “There is no doubt at all that Greeks and Armenians disliked each other and that at times this dislike turned into bitter hostility…”, but also, many Armenians

“were thoroughly integrated into its political and military life, identified themselves with its interest and adopted the principal features of its culture. In brief, like many other elements of different racial origins, as, for instance, Saracens, Slavs and Turks, who had a similar experience, they became Byzantines.”

Veering a little bit into historiography rather than history, there is actually a brand new book about ethnic identity in Byzantium, Romanland by Anthony Kaldellis (Havard University Press, 2019). There is a whole chapter about “Armenian” identity specifically. Kaldellis argues that ethnic identity is an invention of modern historians with modern race-based ideas of nationalism and ethnicity. It’s something that Byzantine Greeks/Romans wouldn’t have thought about, at least not in the same way. So people wouldn’t have considered themselves as “Armenian-Roman” or “Roman-Armenian”, they would either be Armenians who spoke Armenian and were part of the Armenian church (and therefore not really “Roman”), or they would be Armenians who had assimilated into Greek culture (and therefore Romans).