Hey Daniella,
Thanks for getting in touch! I really hope our plugins will be a great fit for your client. 🙂
Yes, it is totally possible to auto-import events from Google Calendar into The Events Calendar. In fact, The Events Calendar has a service baked into it called Event Aggregator that will allow you to do that.
Simply purchase a license for Event Aggregator and enter it on the Events > Settings > Licenses screen. That will unlock the feature to import events from Google Calendar on the Events > Import screen.
Does this help answer your question? Please let me know. 🙂
Cheers!
Geoff