All I had to do was go to my CloudKey controller that's registered in Unifi cloud portal. Once logged in, I went to Settings (icon in the lower left part of the screen), navigate to the Networks tab. Select the LAN to upgrade to the 22.214.171.124 DNS service. There is a setting named DHCP Name Server that is set to Auto. Set it to Manual and enter the two DNS server addresses: 126.96.36.199 and 188.8.131.52.
All that stands between me and the awesome 184.108.40.206 DNS service is a simple reboot of all network devices. Enjoy the new DNS service, although in practice you won't notice all that much. Except improved privacy of course.
Update: Not all traffic is safe yet, because at this moment Unifi products don't use DNS-over-HTTPS (DoH). There is a feature request going to get the Unifi Security Gateway to communicate with DNS servers over DoH if it's supported by the DNS Resolver. 220.127.116.11 does support DoH, but Unifi lacks support at this moment. If you want to use DoH, the "easiest" way is to host your own DNS Server and configure that to talk to 18.104.22.168 over DoH. You can find a guide on Scott Helme's blog.