There is no Cleveland County Fire Department regardless of how it's listed in the DB. In fact, I don't know that there are any county-run fire departments in the state at all. Just doesn't seem to be a thing county governments can do or choose to do in this state.
Outside of the cities of Norman, Moore, and Oklahoma City, there are other municipal FDs in Cleveland Co - Noble, Lexington, and Slaughterville. Beyond that, there are a couple of Fire Protection Districts, taxing bodies that exist to fund a fire dept - Little Axe and Cedar Country are both funded by FPDs.
Now, who dispatches those. First three are easy - the respective police departments dispatch Noble and Moore FDs and OKC FD has their own dispatchers (or did - did they get rid of them?). Norman PD dispatches LAVFD. Noble and Lexington PDs dispatch their respective FDs. Beyond that ... I'm not really sure! I believe that the County SO dispatches one of either Cedar Country or Slaughterville (or both) .. but I am no longer certain!
I did not listen to scanners much for the first 120 days or so of the pandemic but I've picked it up in the last couple weeks. Did a lot of FT8 on 50 MHz and HF, just not scanner stuff. I have my 780XLT dedicated to conventional now and the 996P2 dedicated to the OKC and Norman P25 systems ... If I can figure out the FD radio systems in Cleveland Co, I will post what I find.