Is it possible to stop output from a command after bg?

Here's a solution that actually redirects the output of a command while it is running: https://superuser.com/questions/732503/redirect-stdout-stderr-of-a-background-job-from-console-to-a-log-file

For a solution that is more usable in an every-day scenario of using a terminal, you could do wget -o log http://file & to run wget in the background and write its output to log instead of your terminal. Of course, you won't see any output at all in this case (even if you ran wget in foreground), but you could do tail -f log to look at the output as it grows.

Do you mean that you want to keep wget(or some other cmd) running background without outputing messages to the console? If that's the question, you can try redirection

$ wget http://file > /dev/null

the cmd above will redirect the stdout to /dev/null and you won't see messages except error messages. If you don't want to see error messages either, try this

$ wget http://file 1>/dev/null 2>&1

You can google redirection for more detailed information.