Is there a way I can compute MD5 or SHA hashes on Windows and Linux without the need for a specific application?

md5sum, sha1sum, sha256sum, etc. are all standard in Linux distros. They're part of the required coreutils package on Debian and Ubuntu, so if you've installed one of those, you already have them. In other distros you might have to install them, but I don't really see any way around that besides writing them yourself, which I don't think you want to do.

On Windows, I don't know of any native hashing tools. You'll probably have to install something. You can install Cygwin, which also includes md5sum and sha*sum in its required coreutils package, or for a free GUI tool there's e.g. md5summer.