Welcome to the Community, Simon.
Yes, you can export multiple invoices for one customer in QuickBooks Desktop (QBDT) by running a report through the Reports tab. I'll guide you on how.
- Open your QBDT file.
- Go to the Reports tab.
- Click on Customers & Receivables, Transaction List by Customer.

- From the report's page, go to the Customize button.
- Set the Date to 01/01/2019 to 12/31/2021.

- Go to the Filters column, then filter the Transaction Type to Invoices.

- Select only the customer's name for the Name filter.

- Hit OK.
- Tap the Refresh button to get the information you need.
From there, select the Export tab to export the report. Or, click on the Print or Email button to mail and share the details with your customer.
For more tips while running your reports, you can open this article: Understand reports.
Aside from invoices, you can also learn more about tracking your other customer entries in the future from this link: Get started with customer transaction workflows in QuickBooks Desktop.
If you have any other questions about exporting invoices from QuickBooks, let me know by adding a comment below. I'm always here to help. Have a good one!