Can I use Johannes Brahms' music in my short film?

Yes. The gu’s been dead long enough. Some discussion about this issue

edit: guy’s

His music yes, but not necessarily someone else recording of it.

I’m not sure what country you are in, but you should be safe. Here is a list of copyright laws by country regarding music. http://en.wikipedia.org/wiki/List_of_countries'_copyright_lengths

The rule I have always thought is the life of the composer plus 70 years, but the link gives more detail.

Yes, if you remember to credit him.

Not a copyrighted recording of Brahm’s music.

You need to find a royalty-free performance.

Yes, but you have to arrange for recording it yourself, or get permission to use someone else’s recording. Or, as @jaytkay just said – go royalty-free.