Good question. For adverbs ending in "-ly", like "quickly" or "slowly" or "softly", you use "more" or "most" to indicate an extreme. So, while "quickly" and "quick" are adverbs that mean almost the same thing, they become "more quickly" and "quicker": "He ran more quickly" or "he ran quicker". Likewise, "Please drive more slowly" or "please drive slower" "Please speak more softly" or "please speak softer".

QUICKLIER is incorrect. I think most people would say QUICKER. Technically QUICK is an adjective, not a adverb. If this is a concern you could say MORE QUICKLY.

The first option is correct. Quick -> Quicker ☑ Quick -> Quicklier ✖