What is the full name of the file?

Firefox should not be doing that.

Start Firefox in Safe Mode {web Link} by holding down the <Shift>
(Mac=Options) key, and then starting Firefox. A small dialog should appear. Click Start In Safe Mode (not Refresh).

Is the problem still there?

The file name is backup.session. Next time I close Firefox it is backup2.session, and so on. I started Firefox in safe mode and upon closing Firefox, it did not generate the session file. What does this mean?

Those type files should be saved in the Firefox Profile folder in your user folder. Something is changing that. Did you notice any other files going to the desktop?

It could be the work of one of your add-ons. Type about:addons<enter> in the address bar to open your Add-ons Manager. Hot key; <Control>(Mac=<Command>)<Shift> A)

In the Add-ons Manager, on the left, select Extensions. Disable a few add-ons, then Restart Firefox.

Some added toolbar and anti-virus add-ons are known to cause Firefox issues. Disable All of them.

If the problem continues, disable some more (restarting FF). Continue until the problem is gone. After, you know what group is causing the issue. Re-enable the last group ONE AT A TIME (restarting FF) until the problem returns.

Once you think you found the problem, disable that and re-enable all the others, then restart again. Let us know who the suspect is, detective.

This seems like a lot of work for something that really doesn't cause real problems. If I found an extension or plugin that was causing the problem, that I use (I don't keep any I don't use) what would I do? Maybe I will wait for a Firefox update, or if it becomes annoying enough, switch to another browser. But thanks for your help.

That is your choice.