Kubernetes use private DNS

Is it possible to use a private DNS in Kubernetes? For example, an application needs to connect to an external DB by its hostname. The DNS entry, which resolves the IP, is deposited in a private DNS.

My AKS (Azure Kubernetes Service) is running on version 1.17 which already uses the new coreDNS.

My first try was to use that private DNS like on VM by configuring the /etc/resolve.conf file of the pods:

dnsPolicy: "None"
  dnsConfig:
    nameservers:
      - 10.76.xxx.xxx
      - 10.76.xxx.xxx
    searches:
      - az-q.example.com
    options:
      - name: ndots
        value: "2"

Then I tried to use configmap to adjust the coreDNS:

apiVersion: v1
kind: ConfigMap
metadata:
  name: kube-dns
  namespace: kube-system
data:
  upstreamNameservers: |
    ["10.76.xxx.xxx", "10.76.xxx.xxx"]

But my pod is every time running in an error on deployment:

$ sudo kubectl logs app-homepage-backend-xxxxx -n ingress-nginx
events.js:174
      throw er; // Unhandled 'error' event
      ^
Error: getaddrinfo ENOTFOUND az-q.example.com az-q.example.com:636
    at GetAddrInfoReqWrap.onlookup [as oncomplete] (dns.js:56:26)

What am I missing?