I’m glad you surfaced with that problem. I had exactly the same thing happen when I upgraded to 18.04. screen froze with this sort of patterning and the colors of the current underlying screen.
Any attempt to find drivers through additional drivers or through the terminal method was unsuccessful. Apparently they have removed the proprietary Nvidia driver from the 18.04 repository.
My card is an Nvidia GForce 6200 SE (NForce 430) same family as yours. The only driver that cooperates with it is is the Nvidia legacy binary driver version 304.137 available in versions 17.10 which I believe is out of support as of today. It’s also available in version 16.04, the version I had to downgrade to to get things going.
I still have some minor issues that I’m trying to iron out like missing compositors in mate tweak’s windows manager. The only one available is Marco (software compositor).
Also the Nvidia x server setting tool does not launch and the Displays tab does not detect my monitor where they were both working under nouveau and all compositors were available.
But the good thing is it is stable, no frozen screen gibberish or crashes and consecutive reboots.
Also you’ll notice if you inspect the kernlog in the log file viewer section you’ll see indication towards the end of that file that the nouveau driver is faulting at some memory address just before the crashes occur.
All that said 16.04 with the 304.137 is the only combination that finally worked for me and that the driver should be downloaded directly from the ppa.
The one that shows up from additional drivers is the 304.135 which intermittently froze up my mouse and keyboard.
If your intent on using 18.04 with all it’s bells and whistles a new video card would probably be your best bet.
Otherwise 16.04 with your current hardware and the 304.137 driver should work nicely.
In any case if you’re intent on getting it working under 18.04 and you succeed, please let me know how you did it. I’m by no means a linux expert just some one who has been tangling with this exact problem for the last 2-3 months.
Take care and good luck…