I have set catalog backup to run every time I exit LR. Is there a way to set an option to just do the backup without having to click the "Back up" button? I do not find a setting to do so. This a just minor annoyance but if I want to quit work, quit LR after a good amount of work I sometimes quit the program, walk away from the computer and find the dialog sitting there when I return.
No such feature is in Lightroom.
That backup dialog should come up as soon as you click the X, in Windows, or do a Quit PS Lightroom on a Mac, to close LR. So unless you are faster than a Speeding Bullet I can't see how you are missing that dialog box when it comes up and you are walking away from your computer.