Yes, the error should be stored somewhere. I guess it depends on the server settings, if the webmaster has display_errors=On and logging turned off, the errors aren't going to be found anywhere.
I suppose you could store it in a file that's accessible via the Q2A admin area for example (and easily found in the filesystem if the error means Q2A is not accessible at all). Q2A has a debug option, right? That could be used to display errors and stuff on the site, in the case a user cannot use logging.
Pretty much anywhere you look it's frowned upon to display errors like that to the user as it is confusing. (Also by hiding errors you do get a little security by obscurity.) This is from php.ini:
; This directive controls whether or not and where PHP will output errors,
; notices and warnings too. Error output is very useful during development, but
; it could be very dangerous in production environments. Depending on the code
; which is triggering the error, sensitive information could potentially leak
; out of your application such as database usernames and passwords or worse.
; It's recommended that errors be logged on production servers rather than
; having the errors sent to STDOUT.
display_errors = Off
; Besides displaying errors, PHP can also log errors to locations such as a
; server-specific log, STDERR, or a location specified by the error_log
; directive found below. While errors should not be displayed on productions
; servers they should still be monitored and logging is a great way to do that.
log_errors = On