The correct answer is: quicker.
The word “quick” is an adjective that means “moving or happening with great speed.” The word “quickly” is an adverb that means “in a quick manner.” The word “more quickly” is a comparative adverb that means “in a quicker manner than something else.” The word “more quick” is not a grammatically correct phrase.
In the sentence “He cut my hair quicker today than last time,” the word “quicker” is used to compare the speed at which the barber cut the speaker’s hair on the two occasions. The speaker is saying that the barber cut their hair more quickly today than they did last time.
The other options are not grammatically correct or do not make sense in the context of the sentence.