No, the group picker simply lists groups.
I've written fields that do that, for version 3 of JIRA, so I wouldn't try to use the addons in more recent versions. But they weren't hard, so I'd have a look in the marketplace to see if someone has done an add-on that you could use for it. I believe you could also use the Behaviours functions in the Script Runner add-on to change the behaviour of a standard group-picker.
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