Welcome to the Community space, @Sillie.
There are two ways on how to get proof that the quote is already emailed to your customer. First, by checking the Audit History of the Quote transaction and by viewing a Customer report.
To check the Audit history:
To run the Customer report:
To learn more about customisation in QuickBooks, click this article: Customise reports in QuickBooks Online
You can check these articles, on how to personalise your sales forms and running some financial reports in QuickBooks:
Drop me a comment below if you have any other questions about quotes in QuickBooks Online. I'll be happy to help you out.