This is some what of a complicated question because both civilizations would change their calendars from time to time. For instance, the Greeks had the Olympiad Calendar. It only counted years, and it did so by using the Olympic Games in order to figure out a count of how many years had passed. Hippias of Elis (according to Plutarch) had first established the sequence of Olympic Victors. He placed the starting date at the games of 776 BCE. But this calendar was only used for historical purposes.
Greeks also had a Seasonal calendar, but we probably wouldn’t see it as a true calendar as it didn’t really record years, but noted astronomical phenomena, during a given year, that had been witnessed.
Probably the most famous sort of Greek calendar was the Athenian Civil calendar. It is what they used that is closest to a true calendar in a sense. It regulated the various Athenian festivals during the year. This calendar didn’t use years like we know. They more attached it to the name of a yearly magistrates, or archon, and that was used to see the relation of one year to another.
For Rome, their Roman calendar began with the foundation of the city of Rome in 753 BCE. But that was used more by historians in order to see the number of years from one event to the next. More often then not, the year was given in the same manner as the Athenian calendar. It was associated with some consul. The Julian calendar would also adopt this as well.
It really wasn’t until the Gregorian calendar that you begin having a more standardized calendar. For Greece and Rome, they had multiple calendars. A great source on this is Lynn Hunt and his book, Measuring Time, Making History.
