I found a rock in the Mojave desert and it resembles a tree branch including what appears to be wood grain. Is there some test or observation that can be done at home to conclusively answer my question?
I don't think there is a really conclusive absolute test one can carry out without further preparations. In the case of a permineralised fossil, this would probably require thin sectioning and close examination to see if any cellular structure diagnostic of wood remains. Overall I'd think though that if it looks like wood and has been replaced by minerals, it probably is fossilised wood. However, you say you found it in the desert, which perhaps could mean it is some kind of sub-fossil or mummified wood.
The first thing to do would be the standard mineralogy tests to attempt to find out which mineral(s) replaced the wood.