For many centuries (during the time of the Talmud), Jews used a secular calendar system. Dating based on the 'Creation of the World' only starting in the Medieval period.
From "Comparative Jewish Chronology" by Rabbi Simon Schwab:
In the spirit of the aforesaid, a new light is shed on the strange fact that - soon after Ezra and Nehemia - a new method of counting the years was introduced by our Sages, a method which was retained for well over 1200 years by our people. We are referring to the so called Greek Era. In Seder Olam 30 we are told that "in the exile" we are to write into our documents the date according to מנין שטרות אלפא. The term Minyan Sh'taroth means the "Era of Contracts" and refers to the so-called Seleucid era. This era, also sometimes called minyan yevanim, began on Rosh Hashanah 312-311 BCE after the battle of Gaza and the conquest of the Holy Land by Seleucus Nikator, one of the generals of Alexander the Great. The Seleucid era was in use until the Middle Ages when the familiar terms l'bri'as olam was introduced, or re-introduced, by the latter Gaonim, such as R. Sh'rira (cf Rambam hilchos Gerushin 1:27). There are numerous Gittin (divorce documents) still extant which carry the date according to minyan sh'taroth. We can very well understand the bewilderment of a Sadducee wondering why a non-Jewish date was admitted into the sacred documents (Yaddaim 4:8). For indeed minyan sh'taroth was not a Jewish date. It was employed by a majority of nations in the Near East and of the Mediterranean area for countless generations and still is in use in some Eastern groups.
Finally, the names of the months used by Jews (for many centuries) are based on the Babylonian calendar. They are not "the months that Hashem appointed," and are in fact based on pagan worship. (The Torah doesn't mention names for months; it just has numbers - first month, second month, etc.)