My grandmother informs me that my great-grandfather used to be a sheriff's deputy in California in the 1930s. While out one day (not sure if investigating or just on patrol), his partner was murdered by a man using an axe, with my great-grandfather being present for the event. I do not know any other details of the incident, I only have my great-grandfather's name and the county he worked in.
How do I collect more information on this incident, the events leading to it, and the fallout?
Definitely start by contacting that county's library. Most libraries can do a reference by mail service for a nominal fee and will dig through microfilm of old newspapers to get you started (assuming it would've been covered in a local paper). If it's a pretty rural area you might receive better, more thorough reference services by contacting the nearest major city library (San Diego, Sacramento, San Francisco etc) instead. Basically a good local librarian should be able to tell you what information is out there, though, and point you towards other resources. I'm mostly a lurker in this sub, not a historian, so I hope I'm within the rules responding here. Good luck!
Check the state archive for that region. I'm in a history research class right now and we just took a tour of our state regional archive. They have death records, jailhouse records, court reports, etc. The lady at our archive is incredibly nice and loves helping people research topics. For reference, here is the website for my county's regional archive. It kind of explains what is in the collection. I'm sure you can find where your regional archive is and give them a call. Good luck!
I'd start by looking for a source that can tell you about any in-the-line-of-duty deaths for the agencies in question. Law enforcement agencies tend to be pretty meticulous about keeping these sorts of rosters, then using the date to examine coverage in local newspapers for a few days afterward.
That said, I did a quick-and-dirty Google search, and http://www.odmp.org (I can't vouch for it, but it does list a lot of in-the-line-of-duty deaths), doesn't show any deaths by axe in the 1930s for the state (unless one was mislabeled a stabbing). It does, however, show one death by pickaxe, from 1938.
http://www.odmp.org/officer/7215-officer-willard-johnston
If that's the right case, here's a little bit of aftermath:
A further Google search indicates that the Santa Cruz News might be the paper to look in for full details.
You should be able to get the newspaper coverage from a library in the area as outlined above. The search will just be a lot cheaper/quicker if you know exactly which few days to look for to start with.