Just thought I'd drop my 2cents.. I owned a Nikon d5200 last year for about 6 months before replacing it with a gx7. In April I tested the a6000 against my gx7 and found the a6000 to have no advantages. I became so confident about the Panasonic system that after 7 months of using the gx7 I upgraded to a gh3 (for the higher bitrates, better grip and more spaced out controls). I can't speak for the g6 but I doubt the image quality is much different to the gh3/gx7.
Edit:
Just to expand on a few things...
I had a very hard time leaving my Nikon because the photo quality was just so damn good. The video quality was good too but I always felt it lacked a little something. But there were other problems, like the small viewfinder made it hard to manually focus for stills, especially at night. Changing WB for video was cumbersome. And the lack of dual control dials was a hindrance too.But mainly, there was no cheap and small 35mm focal length equivalent lens (a personal preference).
The a6000 felt nice in the hand and had a great electronic viewfinder. However, like previous NEX's it showed a lot of noise in low light, making it very difficult to manually focus. I did a little stills test and the latitude and noise levels between it and my gx7 were very similar, with the GX7 gaining a lead at high ISO's. The video quality was no less disappointing, with the Sony showing more moire/aliasing, less detail and less dynamic range.