Though not every place on earth has 4 distinct seasons. The tropics have two (wet and dry), and many cultures have/had seasons based on farming practices of their primary crop.
Creating an entirely new calendar would be easiest on an entirely new planet.
I actually gave some consideration to that. There are also places that have six seasons. However, I'm pretty sure that a majority of the world's population lives in temperate zones (I didn't bother researching that, so I could be wrong), which have four seasons.
Even if you decide to go with four seasons, the definitions of the seasons get tricky. You can define them by the solar calendar, or you can define them by environmental conditions; the two tend to be out of phase by about six weeks. I actually figured out a way to make the months line up reasonably well with the four environmental seasons and with the six environmental seasons (which are not of equal length); it wasn't perfect for either, but it was OK for both. I didn't bother keeping notes...
Creating an entirely new calendar would be easiest on an entirely new planet.