It's an apples-to-oranges comparison, really... the GRII and Q7 are very different beasts with different strengths and weaknesses.
The GRII has a large APS-C sensor and superb image quality to match, it's pocketable, and has an excellent, reasonably fast, fixed focal length lens. So long as the field of view from an 18mm lens on APS-C works for you, it'll be an excellent choice. Note that, since the lens is fixed, if you end up with dust or debris on the sensor (rare, but it can happen), the camera will need servicing to remove this.
The Q7 has a small sensor, and whilst image quality is very good in its class - especially if you shoot in raw and you're handy with post-processing skills - it will never match a good APS-C sensor such as that in the GRII, unless reproduced at modest sizes. The interchangeable lenses result in a less-pocketable system (though it's still very compact), but with greater versatility where field of view is concerned. The short focal lengths (required because of the small sensor size) result in significant depth of field at all apertures... not good if you want to separate your subject from the background or foreground, but great if you want everything from near to far in focus. If you should end up with dust or debris on the sensor, it's easy to remove with a dust blower or brush.
So, as I said, different beasts with different plus and minus points. Essentially, though, the GRII wins on absolute image quality and pocketability, while the Q7 wins on field-of-view versatility