Hello! For my field, which is Byzantium, the short answer is... No.

The Anno Domini calendar was not used in the Byzantine world in 666 A.D. It would have been the year 6174–6175 according to their calendar, which dates from their believed creation of the world (anno mundi). As the Byzantines were the preeminent Christian realm of the 7th century, the Christian world would not have been concerned.

As others have noted, the year 666 didn't play a role back then, mostly because the calendar was different. The seventh century was still however a time when apocalyptic expectations reached an all-time high, but it was for very different reasons. Already at the tail-end of the sixth century, important figures like Gregory of Tours in Francia and Pope Gregory the Great in Rome were pointing out that the world was becoming increasingly fractured, where evil deeds and evil men triumphed more often than not, whilst society as a whole was plagued by war and disease. They didn't really put a date on the impending apocalypse, but they thought that it was close and that they needed to do all they can to rally their flock and fellow clergy to prepare for it.

Events in the seventh century would, to a certain extent, vindicate their views. Amidst the great war between Rome and Persia (603-628), Jerusalem was sacked by the Persians in 614, leading to a great deal of soul-searching and lamentation by Christians across the Mediterranean. Imperial propaganda changed tack as well, as the emperor Heraclius began to be portrayed as an almost-Messianic figure, leading his people in a holy war against the Zoroastrian heathens. These expectations were furthered in 626 due to the unprecedented siege of Constantinople and although the Romans would eventually win the war, their hopes of a more peaceful world were once again dashed, as the Arab conquests began in the 630s. For the second time in two decades, the Holy Land was lost and once again people feared that the empire itself would collapse. By the 640s, the imperial government perhaps even commissioned an apocalyptic work, the Vaticinium of Constans, to bolster their claims - it portrayed the emperor Constans II as the legendary 'Last World Emperor' who would unite Christendom and upon victory, herald the beginning of the End Times. The emperor was still a teenager, but the regency government was plagued by internal disputes and external defeats, so perhaps they thought it best to appeal to the popular imagination in an attempt to regain some legitimacy for the emperor.

These traditions continued to mutate as time went on, as can be seen in the work of pseudo-Sebeos, an Armenian historian writing in the late 650s, who thought that the impending Arab civil war would be the end of caliphate, which he thought was the fourth and last kingdom predicted in the Book of Daniel from the Old Testament. He must have been sorely disappointed when the Arabs emerged from their civil war stronger than ever in 661. This victory also meant that the Romans and the Arabs now re-opened their 'Mediterranean world war' with each other, eventually culminating in the Arab siege of Constantinople in 668. Though the Romans didn't know it as the 660s at the time, the decade was indeed a time when eschatological sentiments permeated through society. They didn't die down after this either, as the Apocalypse of Pseudo-Methodius is a great example from the late seventh century; it was a work that expounded on the same themes as the the earlier Vaticinium and eventually became a best-seller across Christendom in the middle ages.

Other interesting things can be pointed out as well, though I am not very knowledgeable about them. Islam for instance is seen by some historians as a religion that was originally an apocalyptic movement that had much in common with the wider intellectual milieu. There are also multiple Jewish Apocalypses from the same time, such as that of Zerubbabel, that played upon the same fears during the Roman wars against Persia and the Arabs. Meanwhile in the West, individuals such as Isidore of Seville were recalculating their calendars and trying to move away from the apocalyptic fears of their predecessors. Even so however, they were still engaging with the same sort of ideas and as worse and worse news continued to flow in from the East, I suspect that even they felt some trepidation, especially as they harboured fears about the Year 6000 (from Creation), though luckily for them there were multiple reckonings in use so there were several possibilities for when that will occur. Bede, the eighth-century historian who would popularise the AD system, also had some thoughts about the apocalypse; I have some suspicions about his links with the wider context, but I haven't read the two works on it yet (Bede and the Future (2014) and the Bede and the End of Time (2012)), so this is just me speculating freely.

The same goes for my other points as well actually. There is a lot of literature on various apocalyptic traditions and I definitely haven't read enough to give you a full (or super-accurate) overview on the sentiments leading up to 666, but I hope they have been helpful nonetheless. For further reading, James Palmer's The Apocalypse in the Early Middle Ages (2014) is a nice up-to-date introduction to this topic.