If you are looking to get notified when something changes in the calendar, have a look here:
https://developers.google.com/google-apps/calendar/v3/push
Note that push notifications will always post the updates to a URL you register, so you must have a server to be able to use this.
Once you get the notification on your server, you can use one of the many "push from web to browser" frameworks (google for them) to get the information to your app.
This question might also be relevant, although you didn't specify what client you are targeting: Using Firebase to send and receive push notification for an iOS app