DNS settings in Linux are generally stored in /etc/resolv.conf file, you could basically just edit this file to change the DNS settings in any Linux systems. However, the change is not permanent, it will be overwritten by Network Manager when u reconnect or reboot. So, to make the change permanent, there are two methods, Network Manager’s dispatcher script method and the immutable flag method.

Using Network Manager’s Dispatcher Method

This solution is based on ArchWiki. Basically, the steps are:

1. Become root and create a new file /etc/resolv.conf.googledns.

2. Edit /etc/resolv.conf.googledns and add the following lines:

nameserver 8.8.8.8
nameserver 8.8.4.4

3. Add the following line to /etc/NetworkManager/dispatcher.d/12-dns_server:

cp -f /etc/resolv.conf.googledns /etc/resolv.conf

4. Add executable bit to the script:

# chmod +x /etc/NetworkManager/dispatcher.d/12-dns_server

5. Restart Network Manager:

# systemctl restart NetworkManager

What these steps do is just to replace /etc/resolv.conf with the file /etc/resolv.conf.googledns everytime you connect.

Immutable Flag Method

You can also try adding immutable flag to /etc/resolv.conf once you have changed the DNS (nameserver field) in /etc/resolv.conf file:

# chattr +i /etc/resolv.conf

Once it’s set, not even the root can modify this file, so you have to remove immutable flag first before editing this file:

# chattr -i /etc/resolv.conf

I have not personally tested this method though, there might be some complaints by NetworkManager that it is not able to write to /etc/resolv.conf.