history - Why use Babylonian calendar instead of our own?

Why did the Hebrews not create their own calendar system instead of copying the pagan Babylonians'?

I find it interesting that they use a pagan calendar which often honours pagan gods. Why is this permitted?

A source for the claim that the Babylonian months honored pagan gods:

Tamuz - Hebrew Month

The name of the month was adopted from the Assyrian-Babylonian calendar, in which the month was named after one of the main Mesopotamian gods, Tammuz. This is referred to in Ezekiel 8:14.