Android to Google calendar sync problems
I am assuming you are on a rooted device.

Use your preferred file explorer to browse to : /system/app

There should be two apk files. One the calendar app (calendar.apk) and the other calendarprovider.apk. The second apk provides proper synchronization between the google account and the actual calendar application which is probably missing or not working. 

You should be able to find a replacement apk for your specific device and rom on XDA.