Yes, you can change the look of the Quick Navigation menu. You should create a whole new theme for several components, which will act a whole for the Docking control:
In Visual Style Builder, go to File >> Load Control >> Telerik.WinControls.Docking. There you will find the four elements which you need for the Docking control.
Make the themes for RadDockableSelector (the part which you are concerning), RadDockingGuides, DockPresenterControl, DocumentPresenterControl. Set their ThemeNamem e.g."Green" (should be the same for all of them).
Currently, we have some issues concerning this theming scenario and here is how to workaround them:
First, set the theme for the Docking at design-time.