Not that I've ever seen. The Romans, at least in the Republic and early principate, wouldn't have thought to do so because they named the years after the presiding consuls instead of counting from an arbitrary date. Other parts of the Roman world did have local traditions, but the only ones I can think of restart their numbering with every monarch (Egypt, Babylon), or use a system similar to the Romans (Athens).

Eventually the Romans did start counting from the founding of the city, which is problematic, but someone who knows Late Antiquity could tell you more about that.

In our modern sense of periodization, almost certainly not. Did the Roman's chunk together periods of their history to understand it a bit better? It's quite possible, we cannot know for sure. Fashion and culture certainly changed, we have busts of prominent romans to show the transitions of fashions and the favored style of art as well. It would be a mistake, however, to profess to know how the everyday Roman judged the world around them and their past history. The definition of a time period in Rome, outside of Consular and Regnal years, would have been totally subjective. The best I can give you is a maybe.