I assume that you are based in the U.S. I'm not really familiar with the U.S. system of "grad school" so take what I say with a grain of salt.

Publishing a paper is certainly a good way to show your professor, that you are capable of doing research but probably not absolutely necessary. Having a reputation as a reliable and capable student should also go a long way to convince your professor that you are a good cancidate.

Working with one of the PhD students on their research project should also be a good way to earn your professor's trust.