Are you looking to push events to Google Calendar or pull events from it?
Event Organiser can push events to Google Calendar for free. It provides a calendar subscription feature that your visitors can use to subscribe to an iCal feed of your events.
If you want to pull events for a Google Calendar, there’s an additional paid plugin that provides this feature:
https://wp-event-organiser.com/extensions/event-organiser-ical-sync/
