Interesting SK, I checked in my labs, on an open server VM R81.20, with DNS configured and the interface eth0 configured with DHCP, and the gateway uses the DNS received from the DHCP.
[Expert@CP-LAB-fw:0]# cat /etc/resolv.conf
; generated by /sbin/dhclient-script
search fritz.box
nameserver 192.168.1.254
[Expert@CP-LAB-fw:0]# nslookup
> www.google.it
Server: 192.168.1.254
Address: 192.168.1.254#53
Non-authoritative answer:
Name: www.google.it
Address: 142.251.140.99
>
CP-LAB-fw> show configuration dns
set dns mode default
set dns suffix fritz.box
set dns primary 8.8.8.8
CP-LAB-fw> show configuration dhcp-client
add dhcp client interface eth0
set dhcp client interface eth0 timeout 60
set dhcp client interface eth0 retry 300
set dhcp client interface eth0 reboot 10