Do you need to quote other historians in exams?

In most A Level mark schemes, this is not a requirement.

Examiners are looking for you to demonstrate your understanding of key historical concepts and ideas, show that you understand the significance of a question (it was asked for a reason), show the ability to explain and analyse those ideas in relation to the question (rather than just narrate or describe), develop your ideas within the essay and write a clearly structured argument which follows for the reader.

None of these requirements demand that you unthinkingly parrott other historians.

However, depending on how you learn, associating key ideas or arguments with particular historians or groups of historians can help you remember the core parts of your course. Also, it is an easy way to show historical knowledge.

Ultimately, if you learn in a way that suits memorising other historians's quotes, then it can be very helpful. But it's not required.