The router has a DNS configured and works with all other devices.
I'm not sure what this means. You mean that you have multiple devices, and you've set multiple of them up to use 192.168.1.1 as a DNS server, and some work and some don't? Or you just mean that the other devices work in that they can access the Internet, and aren't using 192.168.1.1 as a DNS server?
If you've got a Linux system, the basic tools I'd probably use as a first pass are dig, which is a utility that provides a fair bit of information about what's going on, bypasses your system resolver, and is often used for diagnosing DNS issues, and tcpdump, which will let you see what is going by at the network level. On Debian-family systems, dig is in the package bind9-dnsutils, and tcpdump is in the tcpdump package.
In one terminal:
$ sudo tcpdump udp port 53 and host 192.168.1.1
That'll start printing a summary of DNS traffic to and from the Linux box and the router.
In another:
$ dig www.google.com @192.168.1.1
That'll let you see if the query is timing out or what-all is going wrong, and it'll ensure that you don't have some client-side configuration issue, since you aren't using your system resolver.