For your router box you can add in /etc/resolv.conf
so it can use google public dns for resolving.
Now if you want the internal box to use your router as a resolver, you have to setup the router box as a caching/forwarding dns. For this you need to install a name server such as dnsmasq
Use your distro's package manager to install one of them (dnsmasq is more lightweight) and read a howto about configuring a caching name server using the installed package