Strictly speaking we would say "more quickly" but many people say "quicker" these days so I think it has become acceptable.

Fang, you've asked an excellent question. Ruthi answer is correct. There are some adjectives and adverbs that have the same forms, for example: "a fast car goes fast / if you do hard work, you work hard." In other cases, the adverb may have two forms, one like the adjective and the other with -ly, for example: "fast" can mean both quick and quickly / "loud and loudly" He's singing loudly or He's singing loudly. "quick" can be an adjective or adverb, but only in an informal style such as speaking or informal writing. In formal writing, it would be more correct to use the adverb forms you have mentioned, i.e., quick, more quickly, most quickly... crazy English :)