What Threadless adapter should I use? I want to replace the quill stem on my Roadbike with a Threadless head post.   

The measurement shows about .8 inches.
Does a 1 inch adapter mean it is converting to a 1 inch diameter or does it mean the head post diameter should be 1 inch?
For a road bike the quill / threadless adapter size it is almost certain to be 22.2 mm (0.874"). When people talk about a 1" steering tube 1" is the OD, the ID which is the actual size of the stem is usually 22.2 mm.

The description of the adapter will call it a 1" to 1 1/8" or a 22.2 mm to 28.6 mm adapter.

When you install it you don't need the spacers under the stem that would be used in a normal threadless setup. To me the look took some getting used to, but I loved being able to swap handlebars without having to strip everything off one side.