Family duty is not really a term in Quesnel's vocabulary. He's all too happy to let his niece suffer at the hands of Montoni as long as she stays out of his hair. And he's not shedding too many tears over the death of Madame St. Aubert: he and his wife "seemed to have forgotten that they ever had a sister" (1.2.14).
We might actually think that he's a forgetful guy if we didn't already know that he just doesn't care. Yeah, he's quick to forget about Emily—unless she writes about something that really sparks his interest.
You know what's interesting to Quesnel? Money. Like Madame Cheron, Quesnel is interested in getting more of it and using Emily to that very end. He's all too happy to marry her off to Morano, knowing that it's a great monetary match. And he has no qualms about renting out La Vallée to the highest bidder before St. Aubert's coffin is in the ground.