The correct word to use in this sentence is "quickly."

In this sentence, "quickly" is an adverb that describes the manner in which he walks. Adverbs are used to modify verbs, adjectives, or other adverbs. In this case, "quickly" modifies the verb "walks" and tells us how he walks.

The word "quick" is an adjective, which describes a noun or pronoun. It does not fit in this sentence because we are describing the action of walking, not the person.