How to extend Salesforce 'send quote by email' page?

Hi Audrey

Please follow the below steps

1. Create a VF page with Quote as standard controller, and an extension class. The extension class can query all the information you need related to the quote like the related opportunity, account information, etc. Check out this link on how to convert a VF page to a PDF:https://developer.salesforce.com/docs/atlas.en-us.pages.meta/pages/pages_quick_start_renderas_pdf.htm
2. Override the Create PDF standard button with your VF page. In your VF page, you can replicate the functionality of the standard PDF page generated by Salesforce with buttons like "Send Email" for example.
3. You can invoke your 3rd party API to send the email using the button.