Hi Gene kadish,

Migrating the .qat files to Office 365 directly is not allowed. Actually Office 2007 is supported in Office 365, and you can connect Outlook 2007 to your Office 365 mailbox. You can refer to Set Up E-Mail in Outlook 2007 to do this. Please note that it's needed to use Outlook 2007 SP2 or above versions.

In addition, you can choose to use Office Professional plus or Office 2010 as your desktop application, rename the *.qat files to *.officeUI filles, and copy them to your Office 2010 folders.

Thanks,
Reken Liu