Although there may be some, way more easier methods, I would select the mid segment (all 4 edges) and apply a Chamfer command to those edges. That creates two duplicates of the 4 segments at each side of the first. There is one checkbox in the Chamfer interface, which is checked in the image. The second one from the bottom. That checkbox creates an empty area with parallel borders.
Select those borders in Border Sub-object selection mode and Press Cap. (You can see below, Border and Cap and the place of Chamfer, too)
I strongly recommend checking the number of vertices when you're modeling with Edit Poly tools. It won't be doing something it shouldn't, but there are times you may create bad models without knowing where the command adds the new faces or edges. You can also use the StlCheck modifier for errors in a model.
Hope this helps.
I believe single quad face, with edges on a rectangular middle segment as in your model would not be possible unless it is a separate element. Then you can simply attach. You can not weld the selected two vertices in the image. If that is what you are asking.
Although you can create a face, sharing the edge between two adjacent faces with triangular "mesh" model type, the resulting model with 3 adjacent faces will have 3 separate edges on each other, sharing the same two vertices, eventually having the model error, as named in Stl-Check modifier as "Multiple Edge"
