Can GNU Grep output a selected group?

You can use sed for this. On BSD sed:

echo "foo 'bar'" | sed -E "s/.*'([^']+)'.*/\\1/"

Or, without the -E option:

sed "s/.*'\([^']\+\)'.*/\1/"

This doesn't work for multiline input. For that you need:

sed -n "s/.*'\([^']\+\)'.*/\1/p"

While grep can't output a specific group, you can use lookahead and behind assertions to achieve what your after:

echo "foo 'bar'" | grep -Po "(?<=')[^']+(?=')"

You can use \K to reset and discard the left hand match text along with a lookahead which is not added to the match text:

$ echo "foo 'bar'" | grep -oP "'\K[^']+(?=')"
bar

GNU grep only.