Within the context of the story alone, Josiah Quincy is first mentioned in chapter 4, when Johnny is in jail. He is a "famous Whig" attorney who agrees to defend Johnny free of charge. Other than this, he is a relatively minor character in the story.
A few inferences can be made about him based only on clues in the novel. First, he is a Whig, who were known for their revolutionary ideas and willingness to go to violent measures in order to gain the freedoms they believed they deserved. Josiah Quincy is present at the first "secret" meeting which ends in the decision to dump taxed tea into the Boston Harbor (the Boston Tea Party). The fact that he agrees to defend Johnny for free suggests that he is wealthy.
Outside the context of the novel (historically speaking), Josiah Quincy was in fact an attorney, wealthy as a result of his family connections, a congressman, judge, state representative, the mayor of Boston, and the president of Harvard College.