VNC notifications example:

#!/bin/bash

alertme() {
    zenity --notify "Big brother watching"
}

while :
do    
  netstat -a | grep vnc | grep ESTABLISHED && alertme
  sleep 60 #wait 60 seconds
done

Notify-Send script with more options than zenity:

https://github.com/vlevit/notify-send.sh/blob/master/notify-send.sh

./notify-send.sh --expire-time=-1 --icon=/usr/share/icons/Humanity/apps/24/application-x-vnc.svg Subject Message