Similar to most programs, you can change certain behaviors in the settings of the program. For Firefox update behavior:
Settings -> General -> Firefox Updates -> "Check for updates but let you choose when to install them".
Or, you can choose "Automatically install updates" and subsequently check "When Firefox is not running".
Either way should address your issue.
Those settings don’t do anything bec even when I select “Check but don’t install” it still nags me and eventually installs even if corp IT policy is only to update monthly.
I wish there was a way to turn off the nagging too. Why can’t Firefox trust my corp IT to follow their process?
That is definitely possible. In my corporate environment, I don't get any nags or prompts to update, and when I go into settings it shows "Updates disabled by your organization".
It sounds like your IT left the update prompts enabled, perhaps specifically to let you update at your own convenience and hence avoid the problem you complain about when it forces an update while you are using it.