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From what I can gather from your post, what you created is not a Quick Project. A Quick Project is set up in ArcPad. I am assuming that you have the Trimble Positions ArcPad Extension installed on you GPS unit. If you set up a file geodatabase (not a personal) prior to field data collection, you need to gnss enable the geodatabase and gnss enable the map document. This adds a TrimbleSessions feature the file geodatabase and adds that sessions feature to the mxd.
To post-process a Quick Project, you only import/check-in from the device (though I reccommend copying the folder from the device to your computer and checking in from the computer). First, you would open a new blank mxd and open the TP Desktop Administrator window, click on "Projects" then "create">Import ArcPad Quick Project. This creates a new Project. You will have a check box that if checked will create a new file geodatabase to store your Quick Project data and save the map document. Alternatively, you can assign a geodatabase for the data to be imported into and still save the map document.
Once you have imported the Quick Project, the project should appear in the "Projects" list in the TP Desktop Administrator window. Make sure the imported Project is highlighted green and open the rimble Positions Desktop docking window. Your Quick Project should be listed with "Waiting for Correction" next to it. Click on "Correct" and select a processing profile.
Hope this helps