I cannot Quit Firefox on my Mac. This means I can't shut down my computer and I cannot install updates. This is ridiculous.

Shutting down the computer will end the process. You can force a shut-down by pressing and holding the power button for about 5 seconds, note; this is not recommended. Alternatively, there are two other options. What I would recommend is to open the task manager by pressing cmd+alt+esc, then select Firefox and end process. There is also another option, go to ~/Applications/Utilities and open Activity Monitor, from there select Firefox and press Quit Process.