Probably not – as you’re probably talking about biometric unlock. (and not login)
Wrong order… You first have to open the desktop app, and make sure that biometric unlock and “Allow browser integration” are enabled.
Then, with the open desktop app, start your browser and set up biometric unlock on the extension.
For Firefox on MacOS – if it still doesn’t work – have a look into these additional instructions: Firefox browser extension biometrics integration stuck on "Awaiting confirmation from desktop" - #25 by Nail1684
PS: You might as well check the general points for setting up biometric unlock for the extension – this post including the following posts provide the latest summary on that here.