Does DNS server need to be changed in both router and adapter?

So I'm a Verizon FIOS Quantum subscriber. I can easily change to Google's DNS in the Ethernet network adapter, but need I change it in the G-1100 router also?

The pc only uses the ethernet but it does no harm changing on the router and it may be used to check router updates

The preferred configuration would be to configure the DNS servers you want to use upstream on your router and then to configure your client systems either for DHCP which will auto configure the DNS server to point to (preferably the router). This way your router acts as a local DNS cache and forwards the requests to the specified DNS servers on the router if the query isn't already cached.

Configuring this way provides for uniform DNS resolution in your network as well as performance due to the caching.

Technically speaking you can configure any DNS server on any client, but it will have to go out to that DNS server every time a DNS query is performed that isn't cached on the client.