Is there a quick way to look for a specific base?

In general no. If the base is in your alliance then you can ask the Leader / R4 to locate it for you. They can see locations of alliance members.

If not then also ask your alliance to help you find it as they may spot it before you do and can tell you where it is. You can also ask in state chat for help but not if yo are trying to find them without letting them know.