Could the gOS (It links to the gOS's homepage) be the google's operation system?

Click here for more

No, gOS is a crappy Ubuntu derivative. Its literally just ubuntu with a few graphical changes/software additions.
If you want to try a good ubuntu copy cat try Linux Mint. Its probably the closest thing to usable that a linux distro will ever get.

Google's OS is android. Currently its only for phones but there is speculation that it will move onto other devices. If thats true/when that will happen is still up in the air.