Others have correctly posted that you can use $? to get the most recent exit code:
wget_output=$(wget -q "$URL")
if [ $? -ne 0 ]; then
...
This lets you capture both the stdout and the exit code. If you don’t actually care what it prints, you can just test it directly:
if wget -q "$URL"; then
...
And if you want to suppress the output:
if wget -q "$URL" > /dev/null; then
...