What I believe you are referring to is your ZoneFile. This cannot be configured from one vendor to another. For example GoDaddy to DigitalOcean.

You will need to choose to point your domain to the name servers of HostGator or DO. This is not to say when you do, that you cannot point DNS Records at resources on either service.

Example: domain.com is setup with ns1/2/3.digitalocean.com. From within the DO Networking section you can enter a sub domain X that pointed to a droplet in DO and another sub domain Y that pointed to your Hostgator Server.