It’s not so far-fetched. After all, the Greeks learned writing from the Phoenicians, so why couldn’t they have picked up an astronomical calender from the Babylonians? I’m referring to the time line, not the device itself. Few cultures exist in isolation, and the Greeks certainly did not.
Ptolemy absolutely and unequivocally refers to Babylonian observations in Almagest, citing Hipparchus as his source. He also teachs and advocates the Babylonian base 60 number system in the same volume for making calculations, the alternate number systems in use in those days (similar to Roman numbers) were too cumbersome for calculations. The Babylonians kept records of planetary and eclipse observations going back for a couple of millenia by Ptolemy’s time. We are closer in time to Ptolemy than he was to his sources.
The Babylonians never figured out the precession of the equinoxes, something that Hipparchus understood and may have actually discovered. They did know about the variable speed of the sun, basing it on observations of lunar eclipses over many centuries, and correctly deducing that the position of the sun during a lunar eclipse must be opposite the moon, as viewed from the earth.