Yes, ancient culture numbered in some ways their years.

To make some examples: Romans used the AUC (Ab Urbe condita = since the foundation of city) that had its starting point in 753 BC, the year when Rome was founded according to legends. Marcus Terentius Varro, a I century BC historian ( 116–27 BC) calculated that Rome had been founded on the 21 of April 753 BC, according to the fact, claimed by the historian Dionysius of Halicarnassus (60 BC- after 7 BC), who in his text Ῥωμαικὴ ἀρχαιολογία (roughly we can translate in Ancient History of Rome) claimed that the monarchic period of Rome lasted 244 years. Varro calculated that 753 was the year of foundation after establishing that the Republic started in 509 BC (245 AC) with the consuls Brutus and Collatinus. The people of Rome used eponyms from the name of the two consuls to distinguish years, while the AUC was used mostly by historians and emperors who celebrated anniversaries (Philip the Arab in 248 AD celebrated one millenia from the foundation of Rome).

Dionysus of Halicarnassus instead claimed was founded in 750 BC, 437 years after the fall of Troy (many authors, most notably Vergelius linked the origins of Rome to descendant of Troy who fleed from the city during its sack), in the year of the 7th Olympics Games, this brings us to the Greek way to count years, or at least the way some greek historians used.

Timaeus (about 345-250 BC), a greek historian born in Sicily (Most of the cities in western Sicily were Greek colonies) in his book Olympionikai (Olympiads) confronted the years of the Olympics games with the ones Spartans Ephors, Athens arconts and Argos priestesses. Since the first recorded Olympics games where in 776 BC, 776 would be considered the first)year of the first Olympics, 750 the year of the 7th olympics and 753 the 2nd years of the 6th olympics, since Olympics games where held every four year, 752 being the 3rd year of the 6th Olympics and 751 being the 4th year of the 6th Olympics.
This way of counting the years was made popular among Greek scholars by Eratosthenes of Cyrene (276-194 BC), who was an historian, mathematician, astronomist and so on, and the Chief Librarian in the Library of Alexandria

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Hebrews counted the years since the foundation of the universe by God as claimed in the Bible, that was calculated to be in 3760 BC but I don't have many more informations.

I'd say that almost every civilization with astronomical knowledge developed their own calendary (The Mayan one was quite famous in 2012) but is more of a generic statement that a scientific law