I would only capitalize judge if it was used as a title in a phrase, e. g. Judge Lerner, Judge Learned Hand
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.