How to disable all Nagios notifications if a specific alert goes off

You will have to look into the dependencies system of nagios.

See http://nagios.sourceforge.net/docs/3_0/dependencies.html

If you want to avoid warnings for hosts behind the router interface when it is down, you could simply use the "parents" option on the hosts behind it. This tells nagios that whenever the router is unavailable there is no use in checking the stuff behind it.

Adding "parents" to all your hosts has the added benefit that the status map becomes much more useful.

define host{
    use             generic-host
    host_name       cisco_router
    alias           Important router
    address         192.168.1.228
    check_command   check_cisco
}

define host{
    use             generic-host
    host_name       office_printer
    alias           A printer
    address         192.168.1.250
    parents         cisco_router
}