Most likely no but read on.
If the adapter screws on easily
all the way to the point its making contact with the plastic case, then the washer should not be needed.
When you are tightening the adapter onto the radio, if you feel it starting to get fairly tight and harder to turn
Before the adapter starts making contact with the case of the scanner, the washer may be needed. You would probably still see an open gap between the bottom of the adapter and the top of the scanners case around the antenna fitting when this happens.
If this happens, the washer would be added so the adapter is not bottoming out the threads before it is really secure or tight to the case.
Be careful though and only use the washer if you indeed know the adapter is getting tight before it screws all the way on.
The washer looks pretty thick in your picture and may only allow the adapter to screw on the threads a turn or two which would make the threads on the scanner vulnerable to stripping out easily. The center pin may also not connect with the washer in use.
If that washer came with the adapter, the person it came from may have known it goes on too far so they included the washer to compensate.
If you place the washer on first and the adapter still tightens maybe 3.5 turns or more before it starts getting tight, the washer is probably fine to use and is likely needed to prevent the threads of the scanner jack from bottoming out inside the adapter.