resolv.conf is used by the local machine to tell how to resolv names. When running commands on the router the name server pointed out in resolv.conf is used. If the local machine is running a name server resolv.conf should point to localhost.
On my router I have two, but it might be that I create one of them myself.
/etc/resolv.conf lists 127.0.0.1
/tmp/resolv.conf (which is referred to from dnsmasq.conf) lists google name servers
The reason that it works on computers on the LAN but not the router most likely is that the computers on the LAN use the name server they get by DHCP (which is the routers address, thus they ask dnsmasq). But the router itself uses /etc/resolv.conf and if it points to external name servers rather than localhost you get the situation described.
It seems as if you misread the question?