The correct answer is "know"
Hector says he does not know when the movie starts.
The word "no" is incorrect since the sentence needs a verb after the auxilary verb "does".
The word "now" is incorrect since it is an adverb which means "at this moment".
The option "knows" is incorrect since the auxiliary verb "does" is carrying the 3rd person conjugation. In this case, the main verb does not take the "s".