How to change my other DNS server to Google DNS for my website?
If you actually host the server you can change your DNS settings at /etc/resolv.conf (Linux). You can add Google DNS there (Primary: 18.104.22.168, Secondary: 22.214.171.124). I use OpenDNS for my website however it's rather low traffic. I imagine on a high traffic website, Google (or any other competent free DNS service) would disallow connections from your server if the requests got too high.
If your using a shared host, you most likely won't be able to change your DNS settings. You can typically only changed the DNS server of a dedicated, virtual and self-hosted servers. Your host will usually have their own DNS server which you are forced to use.
I'm nearly 100% sure that this not what Google DNS does. You don't change your website - Google DNS is simply an alternative DNS provider for home users. It still syncs to the same higher level DNS servers everyone else does.
However, if you *are* looking to speed up DNS requests to your website, I recommend the free cloudflare service (www.cloudflare.com), though you'll need to adjust your domain nameservers to point at cloudflare when requested...