In order to export Purchases to Quickbooks from Briefcase you should ensure that the supplier used in Briefcase is setup as a contact in Quickbooks and also that the name between Briefcase and Quickbooks is the same.
You should also ensure that you have integrated Briefcase with Quickbooks, for more information regarding this review the Quickbooks Accounting integration article.
To batch export purchases from Briefcase to your accounting system you can go to the 'Purchases' tab. You can then use the filter menu to to filter by 'On Order' or 'Received' status purchases. You can also sort/group purchases to filter out the ones you do not want to export. Once you're ready, press the 'Click here to export * purchases" link.
If you would prefer to only export out individual exports, you can do these by viewing the purchase you would like to export, press the action menu and selecting option 'Export Purchase'.