You don't have
to use Google's name servers. You can use any public name server you want. The Google article shows you the method; the user can simply substitute whatever IP addresses (s)he wants for the name servers.
Please read carefully the Linux section in the Google article. If your installation uses NetworkManager then you can't do it via the command line unless you disable or un-install NetworkManager and set up networking manually (see Part 4 of the Gentoo Handbook for the way of doing that if you are using the OpenRC init system, but I have no idea how to set up networking manually if you are using the systemd init system, as I don't use systemd).
If your distribution doesn't use Network Manager, your DNS settings are specified in /etc/resolv.conf.
If your installation uses NetworkManager, NetworkManager will overwrite any edits you make to /etc/resolv.conf. This is because NetworkManager does what its name suggests: it manages networking and will use the parameters configured via its networking GUI front-end. If you want to stop NetworkManager from doing that, you would either have to change the file's attributes (which is not advisable):
or you would have to use NetworkManagerDispatcher to add lines indirectly to /etc/resolv.conf (you can see an example of NetworkManagerDispatcher being used in such a way in my blog post ‘Server not found’ by browser at launch
If you can't enter parameters via the networking GUI front-end in MATE, something is not right. Obviously the purpose of the GUI front-end is to enable users to configure networking. I don't use MATE but, according to the SL package list for the ISO, the front-end used for NetworkManager is gnome-extra/nm-applet-gtk2. Is nm-applet running? You can check via the command line or by using e.g. top or htop. You could also have a look in the file /etc/NetworkManager/NetworkManager.conf and change "managed=false" to "managed=true" (if the former) and reboot or restart NetworkManager, to see if that has an effect.
Honeyman wrote:BTW: Is there a command which allows me to directly check which name servers my system uses?
If that just shows the IP address of your home router, then the router will be resolving names using the name server specified in the router itself. You could of course use your router's GUI (usually via a Web browser) to specify different name servers. If you had a Belkin router, for example, you would use the procedure given in Changing Belkin Router DNS servers
In fact, why don't you just do that? By configuring the name servers in the router, you would avoid having to do anything on your computer.