Few options off the top of my head:

• Open a terminal (e.g. Ctrl+Alt+T) and type “firefox -p &”. The & operator runs the process in the background so it will continue to run even when the terminal is closed

OR

• Use your desktops equivalent to windows “run”. So for example, on KDE use Krunner (Alt+F2 or Alt+Space usually launches it) and type in “firefox -p”; it usually defaults to running a command. There is also a dedicated “Run Command” plasmoid that can be added to your desktop. On Gnome, I think the “run a command” dialogue will do the same (also Alt+F2 I believe).

OR

• Add an app entry to your desktops menu for Firefox Profile Manager. On KDE if you type Profile, “Profile Manager - Firefox” already exists as a Krunner action; so you can easily get it from your menu or krunner just typing Profile. If it doesn’t exist then you can use your desktop’s menu editor to copy the firefox entry and add the -p as the command line argument. On KDE that done most easily by right clicking on the menu icon and selecting “edit applications…” or search for menu editor. Other desktops will be very similar.

If you don’t want to fiddle with a terminal at all you can just make a keybind/shortcut. Depends on your DE or WM as to how to do that.

In the terminal session you launched firefox from:

Ctrl-Z  # Temporarily suspend process
bg      # Put that process into the background
exit    # Cleanly close the terminal session

Or launch Firefox (or whatever application) with a ’&’ at the end to put it in the background from the start.

if you want to launch a thing from X/wayland, maybe consider getting a run box. there are plenty of options. your desktop environment may already have one.

surround it by nohup and put & at the end of the command.

it would be better to run the command using the key combo of alt + f2 and then putting the command there instead of using a terminal window so that you don’t have to bother with nohup.

I’m not familiar with the -p flag on Firefox but in general, adding & to the end of your command backgrounds it and disown detaches it from your shell so it doesn’t close.

firefox -p &; disown should do what you’re looking for, if I’m understanding you right

If you use kde, you can search for “profile manager”, and it will show up, and can be launched from the app menu.

At least works for me. Before this was added, the KDE search/app menu also lets you run commands directly, so I would just run firefox -p in there. No need for a terminal.

Instead of disowning use exec or xdg-open

Why did the other commenters parse the title as a question? I expected directions from OP.