Do references and quotes improve the quality and make an essay sound better in undergraduate and graduate exams, contests, and other such situations? Could you please provide me with a book, paper, or quote that discusses this topic?
This depends on the nature of the essay. For creative writing, references are usually not needed. But for something intended to be authoritative and factual, they probably are. Otherwise your essay will seem like an opinion piece. And even opinion pieces may need references for context.
So, context matters.