Two (old) tricks for the price of one. Well, since it takes two to Tango anyway.
The first is to make your window movable by dragging, especially if the window does not have a title bar (remember the classic WinAmp?) or use a custom-drawn one (Google Chrome is the prominent example here). To make the challenge more difficult, no change to the main window code is allowed. To reveal the secret (which is a matter of using event filtering properly), go to my Qt Labs blog entry: Moving top-level window by dragging.
The second is about Google Suggest. Almost all browsers have supported it. For a minimalistic example on how to do it, check out my other Qt Labs blog on Google Suggest made easy. The demo launches your default browser with the chosen search term, with or without autocompletion.
(It is evident that my default browser is Konqueror. I told you I am not that biased :-)
Now just like last time, who is going to turn this into a plasmoid?