Both are correct and easily interpreted, and the overlap is near-total. But your teacher is right about the shades of meaning: "quickly" faintly suggests that he's literally quick about borrowing, using, and returning the bike, whereas "soon" only references the time and says nothing of the manner.
This does not mean we wouldn't say either one just as readily. We would. But the mental picture would be slightly different.
Note that in past tense we might be more inclined to say "quickly", since "soon" is not available and the alternatives are a little more marked: "soon after that" or "shortly afterwards" or "before long".
Also, in colloquial speech, I'd usually say "fast" instead of "quickly" — but it would otherwise fill the same role described above.