Ooohhh nooo. This is one of those cases where you might not see much benefit here. I just did a search of the G3 to get a look at the chassis, it looks like a casualty of a manufacturer over-packing hardware into a shell. If I'm not mistaken, all of the outgoing air is going through a slit on the bottom-rear of the computer, correct? If so, the problem is likely more-so related to the computer being unable to actually kick heat out fast enough.
That said, you might potentially get some benefit, but I wouldn't say chances are good.
Flipside to this, I have an Alienware 15R3 with an i7 7700HQ and a 1060OC - I was able to re-paste with thermal grizzley conductonaut liquid metal and saw significant drops, but the cooling system on my laptop has massive amounts of venting on the sides, rear and bottom. In your case, new paste might help pull heat away from the CPU / GPU but then you face the challenge of actually dissipating said heat fast enough.
A good way to check if it would help at all, if you are fairly adept at opening laptops, is to try booting it up and run some benchmarks or games with the shell partially removed to allow greater airflow from the fans. If that makes significant difference, that's probably your issue. If it doesn't, it could very well be a paste quality problem.
I hope this helps you figure out how to tackle the heat issue!