Go to ~/.docker/machine/machines/default/config.json and add your own DNS server into HostOptions/EngineOptions/Dns and restart docker machine.
{  
   "HostOptions": {
        "Driver": "",
        "Memory": 0,
        "Disk": 0,
        "EngineOptions": {
            "ArbitraryFlags": [],
            "Dns": ["192.168.99.1","8.8.8.8","8.8.4.4"], <-- set it here
            "GraphDir": ""
        }
}
 
 
Edit:
The Dns setting in config.json seems to be ignored in the new version of docker-machine. The only thing that seems to work is to add line (edit this to match your needs)
"${DOCKER_MACHINE}" ssh "${VM}" "sudo sed -i.bkp '/--label provider=virtualbox/a --dns 8.8.8.8\\\n--dns 8.8.4.4' /var/lib/boot2docker/profile && sudo /etc/init.d/docker restart"
after
yes | "${DOCKER_MACHINE}" regenerate-certs "${VM}"
in script start.sh, which is run every time Docker Quickstart Terminal is started.
Then shut down the machine (if it's running) and open a new instance of the Docker Quickstart Terminal.