Maybe best to open a ticket then
Thanks all for your help and suggestions 
Any update on this, I’m having the same issue on my Qnap running QtsHero (latest firmware)? It’s very strange, I have DNS on my firewall, and I can ping the IP address and Internet IPs just fine, but DNS resolution stopped working a few days ago based on the logs/notifications. And just the same as the OP, access to the NAS shares is working just fine (Qfile, SMB, SSH).
The issue is resolved. I raised a ticket for QNAP support and they suggested installing the latest firmware which seemed to fix the issue, but I did notice that I had multiple static leases for each of the four network adapters on my TS-453 Pro, each lease mapped to the same host name (only one would ever be in use at a time so didnt think it’d be an issue) but after changing it so that each static host name was unique, I didn’t see any more issues.