I have gone through all of the steps to set up the integration with my google calendar and Zoom, but it isn't working. Shouldn't I get a calendar invite with Zoom information? Or shouldn't the client receive this as well?

Hello Brandi. You may integrate zoom and add short code of the zoom join link to the automated notification. Through this auto generated zoom join link will be forwarded to the client. Similarly, you may send zoom host link short code to the employee. Client and Employee can access the zoom link and start meeting directly from their personal panels. You can integrate google calendar just to let clients book appointments. They or you wont be able to generate any zoom link from there.