@trepinator,
I would only capitalize judge if it was used as a title in a phrase, e. g.
Judge Lerner, Judge Learned Hand, etc.
And in the United States, this 'judge's intern' you're writing about it a lot more likely to be known as a
law clerk. And not capitalized; it is not a title. You would write
Judge Learned Hand's law clerk, Jane Smith. And then afterwards just
Ms. Smith when referring to her.