He cut my hair . . . . . . . . today than last time.

Option A: more quickly
This is the correct option. "More quickly" is the comparative form of the adverb "quickly," which compares the speed of cutting hair today with the speed of cutting hair last time.

Option B: more quick
"More quick" is grammatically incorrect. The correct comparative form of "quick" is "quicker."

Option C: quick
"Quick" is an adjective, not an adverb, and does not fit the sentence structure.

Option D: quicker
"Quicker" is the comparative form of "quick," but it should be preceded by "more" to form a comparative phrase "more quickly" in this context.

In conclusion, the correct answer is Option A: more quickly, as it correctly completes the comparative statement about the speed of hair cutting.