On the English language quiz I wrote the sentence "I get tired quickly." And then I was corrected and said that "quickly" doesn't suit here. Instead of it I am to use the word "easily". Can I really use "quickly" in this sentence if I mean that it doesn't take me much time to get tired?
Yes, "I get tired quickly" is perfectly good English, as is "I get tired easily." Each has a slightly different meaning, so the best choice would depend on your intended meaning.
"I get tired quickly" emphasizes the amount of time that it takes for you to get tired, perhaps while engaged in a specific activity mentioned in the context, such as walking or reading.
"I get tired easily" emphasizes the amount of effort, so you might not get tired for a very long time if you aren't exerting yourself. Maybe you could read comics for hours but reading research papers exhausts you after just a couple paragraphs.
By the way, a couple quick Google searches bring up 122,000 hits for "get tired easily" and 607,000 hits for "get tired quickly". This is not a reliable method for making a word choice, but it does suggest that both phrases are normal English. Often Google Books finds good examples of real usage in real contexts.