This isn't possible in the framework currently.
If you're comfortable altering a core file, this can be done though.
In QImageControlBase, in RenderImage, after imagefilledrectangle is called (line 433 in my copy), you can call:
imagecolortransparent($objFinalImage, $clrBackground);
That's it. Now, any image you upload will have a transparent background.
Note that if your image contains the same color as the background, that portion of the image may be transparent as well. To avoid this, set your image control's BackColor property to something that doesn't show up in the image itself.
Having this behavior controlled by a boolean property of QImageControl would be a worthwhile addition to the framework, I think.