If I host a Perforce server on Linux, do I need UE4 installed on that server?

No, you don’t need to have UE4 installed on the same computer.

I am planning to host a perforce server on my old linux laptop at home. My game development team is going to connect to the laptop remotely so we can all access our game files and have version control.

Would my Linux laptop need UE4 and Perforce installed in order for us to use Perforce as version control and a shared file system for my team? Or would I only need Perforce?