The official image provides a way to run arbitrary SQL and shell scripts after the DB is initialized by putting them into the /docker-entrypoint-initdb.d/ directory. This script:
ALTER SYSTEM SET max_connections = 500;
will let us change the maximum connection limit. Note that the postgres server will be restarted after the initializing scripts are run, so even settings like max_connections that require a restart will go into effect when your container starts for the first time.
How you attach this script to the docker container depends on how you are starting it:
Docker
Save the SQL script to a file max_conns.sql, then use it as a volume:
docker run -it -v $PWD/max_conns.sql:/docker-entrypoint-initdb.d/max_conns.sql postgres
Docker Compose
With docker compose, save the SQL script to a file max_conns.sql next to your docker-compose.yaml, and than reference it:
version: '3'
services:
  db:
    image: postgres:latest
    volumes:
      - ./max_conns.sql:/docker-entrypoint-initdb.d/max_conns.sql
Kubernetes
With kubernetes, you will need to create a configmap for the script:
kind: ConfigMap 
apiVersion: v1 
metadata:
  name: max-conns
data:
  max_conns.sql: "ALTER SYSTEM SET max_connections = 500;"
And then use it with a container:
apiVersion: apps/v1
kind: Deployment
metadata:
  name: postgres-example
spec:
  template:
    spec:
      containers:
        - name: postgres
          image: postgres:latest
          volumeMounts:
          - name: max-conns
            mountPath: /docker-entrypoint-initdb.d
      volumes:
        - name: max-conns
          configMap:
            name: max-conns