One of the things about laptops is that they are not made to upgrade the display. The only way that I could see it being possible to upgrade your display on your laptop is if the laptop comes in many different versions and you can purchase the part for a different version of the specific model number of laptop, then you'd have to find someone to change out the display, which on a laptop (particularly a gaming laptop) can be a big job.
You've got it right about G-sync though... Your graphics card is g-sync compatible. So if you hook it up to an external monitor that is a G-sync monitor you can take advantage of G-sync. You are also right about it requiring a special circuit board that is part of the display in order for you to use the G-sync feature that your graphics card (in your laptop) can provide.
In general G-sync doesn't make your display 'better'. It makes the screen tearing when the frame rate drops way below 60fps not as bad. So unless your display is regularly dropping down into frame rates lower than 30, you might not even see any difference.
Most people would prefer a 'better display' to be an IPS panel (better color, usually better brightness, better viewing angle) and extremely high frame rate (120 fps or better through overclocking), and to have a lot more pixels (2560×1440 or 3840×2160 resolution) G-sync doesn't do any of that.
In general, I'd be looking at getting a better gaming laptop instead of trying to 'upgrade the display'. You will probably find that upgrading the display isn't possible.
Buy a better laptop. One that has the features you want built in to it.
Good luck