Mr. Quinion manages the accountant portion of Mr. Murdstone's business, a wine merchant called Murdstone and Grinby. After Mrs. Copperfield dies, Mr. Murdstone confronts David with Mr. Quinion in tow. He informs David that his stubbornness must be destroyed; to achieve this goal, Mr. Murdstone is going to send David to work. The person he entrusts David to is Mr. Quinion. Beyond paying David his weekly wages on Saturdays and finding David a room to rent with the Micawbers, Mr. Quinion doesn't appear in the novel much. He certainly doesn't stand in as any kind of guardian or father figure.
Before this showdown between David and Mr. Murdstone, witnessed by Mr. Quinion, Mr. Quinion provides an early foil for Mr. Murdstone's character. Before Mr. Murdstone marries Mrs. Copperfield and stops pretending to like David, Mr. Murdstone takes David on an outing to nearby Lowestoft. There, David meets Mr. Quinion for the first time.
Even as a child, David notices that Mr. Murdstone is "graver and steadier" (2.84) than Mr. Quinion or his friend, Mr. Passnidge. Whenever Mr. Quinion talks, he keeps looking sideways at Mr. Murdstone to check and make sure that what he's saying is OK. And when Mr. Passnidge tells too many jokes, Mr. Quinion steps in and stops him. The cautious reaction of Mr. Murdstone's friends at this stage of the novel foreshadows what an authoritarian bully Mr. Murdstone is going to be to David.
Mr. Quinion's one other claim to fame is a joke on David early on in the novel, before Mr. Murdstone marries Mrs. Copperfield. Mr. Murdstone describes his wooing of Mrs. Copperfield with his two friends, Mr. Quinion and Mr. Passnidge. They all discuss how Mr. Murdstone is getting along with the pretty little widow and her "encumbrance" (2.73; an encumbrance is a burden) – i.e., her son. And they do this right in front of David's face without David ever knowing, because they refer to David as "Brooks of Sheffield" (2.77). David is too young to catch on, which makes it all the more hilarious for these guys; they finally manage to make David drink toasts to "Brooks of Sheffield" while he remains clueless that they mean David himself.
This just goes to show how naive David is – and also what a difference there is between the David telling the story and the David starring in the story. It's only in retrospect that David understands why these men were laughing at him and thinks the joke worth repeating.