Hi Peter,
That's indeed possible, you'll basically need to configure the router as a DNS server, configure the other DNS server forwarders, and then configure all the local hosts names:
ip domain name example.com
ip dns server ! The router will now act as a DNS server
!
ip name-server 192.168.2.120 192.168.2.121 ! The other server forwarders
!
! Start configuring your hosts:
!
ip host user1.example.com 192.168.201.5 192.168.201.6
ip host user2.example.com 192.168.201.7
! [...]
!
! To configure the router as authoritative for a zone
ip dns primary example.com soa ns1.example.com mb1.example.com
This documentation explains the steps more in details: