Hey there @Jeané46623 - sorry to hear you are running into issues with Google Calendar and Calendly!

The first thing that might be the issue here - you mentioned syncing your business email address with Calendly but not your actual calendar. If you are using your business email address to login to Calendly, that will not automatically result in the associated calendar being connected. You’ll have to set that up separately (using the same email address as desired!). 

To do so:  

Navigate to your Calendar Connection page.

Select which type of calendar you will be connecting. Calendly integrates with the following platforms:
Google - your option!
Office 365 or Outlook.com
Exchange
 
Once connected, take a look at the “check for conflicts” and “add to calendar” settings on your Calendar Connection page.

If you have connected more than one calendar, use “edit” to choose which you would like Calendly to check for conflicts. Any events labeled “busy” in the selected calendars will be read as conflicts, and those times will be removed from your scheduling page.
You can also select which calendar you would like your Calendly events added to. When an invitee chooses an available time to meet with you, the details will be published to your connected calendar and your “Scheduled Events” tab automatically.
 
Note:
You can only publish events to one connected calendar.

The second possibility here - when you say that it says your calendar says it’s connected (meaning you head to your Calendar Connection page and your Google Calendar appears there) but it’s “not showing in the availability section,” do you mean that “busy” events on your Google Calendar are not being blocked off on your Calendly booking page(s)? 

If so, check out this info!

Calendly will view your Google events with this "busy" event status as scheduling conflicts and will automatically remove the time from your scheduling page (including all-day or multi-day events). You can access and change this setting for events already on your Google Calendar by double-clicking on the event title in your calendar and changing the status from "free" to "busy" as shown below:

That should help, first and foremost! Additionally - head to your Calendar Connection page and make sure that your “add to calendar” is the appropriate sub calendar by clicking “edit” next to that section, then do the same with your “check for conflicts” section. If more than one sub calendar appears under “check for conflicts” make sure the calendar(s) you wish to have read for those “busy” events are all selected.

I hope this helps! Let me know if you get stuck and I’ll be back to assist further!