It depends on the analysis.
You could say that quarrel is an intransitive verb and that 'with Joel' is an adjunct. But you could also say that quarrel is a transitive verb that happens to take an argument introduced by the preposition with, so the PP would be a complement. There may be reasons for either position. Which one is "right" depends on the goals of your theoretical approach and the tests available to you to decide.
So that means that what really matters is how your book/instructor/class is introducing adjuncts vs. complements. It's much more important that you can explain why you're choosing one or the other (or even arguments for and against both analyses) based on the tools you have available. And different books/instructors might have different approaches.
I don't mean to give you a non-answer here, and I'd be happy to discuss some of the more abstract theoretical issues, but from the way you've phrased the question you want the 'right' answer for your class. The problem with that is that it actually depends on your class, and also that we won't do homework for you. We can still discuss related issues in general, but it's very important to know that it may not match up to exactly what your book says, so you might actually get an answer 'wrong' on the test if you follow my advice for example.
