Adverbs in comparative clauses

Quicker is the comparative of the adjective quick, so it isn't an adverb and should not, strictly speaking, be used with a verb.

It's true that adverbs ending in -ly don't take the comparative ending -er, so we have to say more quickly. We can say faster but not quicklier.

[1] I run quicker than Sara.

[2] I run more quickly than Sara.

[3] I run faster than Sara.

The usual adverb form has ly suffixation to give "quickly" as in [2], but there is also an adverb "quick", which is formed by conversion from the adjective "quick", and has the inflected forms "quicker" and "quickest". The adverb base form "quick", as in *"I run quick", is unacceptable for most speakers, but the inflected forms are more acceptable than the plain form without ly, which explains why [1] is acceptable for many speakers.