Migration Manager for Active Directory is for migrating objects to either another on-prem AD or Office 365. There is no way from within the tool to backup an Office 365 account. You would get a better answer by asking within a Microsoft Office 365 Community Forum that is available by Microsoft.
Recovery for Azure Active Directory will back up your Account but not the mailbox. If this is what you are looking for you can post your question in the Quest On Demand community at www.quest.com/.../