No. You can use either a Fedora distro or regular default vanilla Ubuntu. Both of these package managers have a special shim keys that are signed by a 3rd party program from Microsoft.
If you want to run anything else, you need to self sign your key for secure boot. Gentoo has killer documentation on how to do this. It doesn't matter what distro you use. Secure Boot is outside of the Linux kernel. With Fedora, it is handled by their Anaconda system, (no relationship to the Python containers system by the same name).