Hello there, @felicia-ray.
I've got some insights about syncing your QuickBooks Online customers with Google Contacts.
QuickBooks Online allows you to import contacts from your Google account. You'll just need to select on Connect your Gmail account when creating an invoice, or import them as an Excel file.
To do that:
- Click on the +New icon and
- Select on Invoice.
- Click the drop-down arrow in the Choose a customer field, then Add New.
- Select on Connect your Gmail account.

If you want to import them using an Excel file, you need need to save it as a CSV. Please refer to the How to export contacts from Gmail section of this link for more details: Import customers or vendors from email contacts to QuickBooks Online.
When it comes to syncing the contacts from QBO to your Google account, this option is currently unavailable. You may consider checking a third-party application that supports this functionality.
Lastly, I'm also adding here some articles that provides information about sending invoices from your Gmail and connecting it to your company:
I'll make sure you're taken care of, if there's anything else that you need with QBO. More power to your business, @felicia-ray.