Yes. Jellyfin is free software built by volunteers and Linux is its main home. The server installs on Ubuntu, Debian, Mint and Fedora, and your existing media library carries across as it is.
Jellyfin by Jellyfin Project · Video · Has a Linux version
Jellyfin's download page lists server builds for Linux, Docker, Windows and macOS, and its own
documentation is written around Debian and Ubuntu first. There is also a Flathub entry,
org.jellyfin.JellyfinServer, which packages the server and its web interface together. It is
released under the GPL version 2 with, in the project's own words, no premium licences or features.
If you already run Jellyfin on Windows, moving it is mostly a matter of copying files. Your movies
and shows are ordinary video files in ordinary folders, and Jellyfin reads them the same way on
either system. The server keeps its own database of watch history, artwork and users, and that can
be copied across too, though it is worth checking the project's migration notes first.
Your phones, TVs and streaming sticks do not care which system the server runs on. They talk to it
over your home network, so the apps on those devices keep working once the new server is up.
One practical note. Playing a file on a TV sometimes means the server has to convert it on the fly,
and that leans on your computer's graphics chip. Which conversions can be offloaded depends on your
hardware and drivers, so test a few files on the devices you actually use before you retire the
Windows machine.