Yes, that is possible. Sometimes the router (default gateway) does DNS proxy lookup on behalf of the LAN clients. The router is not a full DNS server but does cache some DNS information to make lookups faster for the LAN.
You do not necessarily have to configure the DNS server on the router as it might be using a dynamic address assignment (DHCP) from the provider, then the DNS server information will be provided as part of the address assignment. If the address is statically assigned, then the provider may give you a DNS server address on their network to place in the configuration.
If the DNS server is down, name resolution will not work. That is why a minimum of two DNS servers are listed in client configurations. This provides redundancy if one goes down.
See the router manufacturer for specifics on how the DNS configuration is done.