The purpose of Jesus asking the question was not to ask what they did or did not believe about John's preaching, or even his affirmation of Jesus as the messiah. What Jesus was doing was; knowing that they were trying to hold on to their positions as determining what was or was not of God, he was publicly humbling them.
In asking them if the Baptism of John was from God or from man; he was publicly showing that they were no more qualified to determine God's intent than any other person, He was further humbling them by implying what makes you think you can determine the mind of God, but I cannot; in saying "I also will ask you one thing, which if you tell Me, I likewise will tell you by what authority I do these things:". And thereby placing them in a position for him to demonstrate that he was indeed the son of God.
In reading what they were reasoning:
Matthew 21:25 through 27 The baptism of John—where was it from? From heaven or from men?" And they reasoned among themselves, saying, "If we say, 'From heaven,' He will say to us, 'Why then did you not believe him?' 26 But if we say, 'From men,' we fear the multitude, for all count John as a prophet." 27 So they answered Jesus and said, "We do not know." And He said to them, "Neither will I tell you by what authority I do these things.
It is obvious that they understood the trap Jesus had set for them.
The best thing about this Chapter is that Jesus went on in the following verses to further humble them.
Matthew 21:28 through 32 "But what do you think? A man had two sons, and he came to the first and said, 'Son, go, work today in my vineyard.' 29 He answered and said, 'I will not,' but afterward he regretted it and went. 30 Then he came to the second and said likewise. And he answered and said, 'I go, sir,' but he did not go. 31 Which of the two did the will of his father?" They said to Him, "The first." Jesus said to them, "Assuredly, I say to you that tax collectors and harlots enter the kingdom of God before you. 32 For John came to you in the way of righteousness, and you did not believe him; but tax collectors and harlots believed him; and when you saw it, you did not afterward relent and believe him.
The humbling nature of this parable is two fold in that Jesus first berates them for not even being able to understand what the most debased in their society could understand, but went on to berate them for not even believing after those miscreants had showed them.
But Jesus wasn't through yet and gave them a second parable to consider:
Matthew 21:33 through 41 "Hear another parable: There was a certain landowner who planted a vineyard and set a hedge around it, dug a winepress in it and built a tower. And he leased it to vinedressers and went into a far country. 34 Now when vintage-time drew near, he sent his servants to the vinedressers, that they might receive its fruit.
35 And the vinedressers took his servants, beat one, killed one, and stoned another. 36 Again he sent other servants, more than the first, and they did likewise to them.
37 Then last of all he sent his son to them, saying, 'They will respect my son.'
38 But when the vinedressers saw the son, they said among themselves, 'This is the heir. Come, let us kill him and seize his inheritance.' 39 So they took him and cast him out of the vineyard and killed him. 40 "Therefore, when the owner of the vineyard comes, what will he do to those vinedressers?" 41 They said to Him, "He will destroy those wicked men miserably, and lease his vineyard to other vinedressers who will render to him the fruits in their seasons."
The true beauty of this humbling lies in the fact that Jesus has caused them to admit even before his crucifixion that that is God's plan of salvation, and their part in it!
And the final humbling comes in his:
Matthew 21:42 through 44 Jesus said to them, "Have you never read in the Scriptures: 'THE STONE WHICH THE BUILDERS REJECTED HAS BECOME THE CHIEF CORNERSTONE. THIS WAS THE LORD'S DOING, AND IT IS MARVELOUS IN OUR EYES' ? 43 "Therefore I say to you, the kingdom of God will be taken from you and given to a nation bearing the fruits of it. 44 And whoever falls on this stone will be broken; but on whomever it falls, it will grind him to powder."
Jesus is telling them that they will soon not be the ones who dictate the way into the Kingdom of God, and warns them that when it happens those who repent and accept him as Savior will fall from their lofty perch, but that those who refuse to accept God's grace are destined for destruction.
and the chapter ends with their realizing that he had just chopped them down to size:
Matthew 21:45 Now when the chief priests and Pharisees heard His parables, they perceived that He was speaking of them.
Since you desired an answer from the book of Matthew it seemed that the answer was in the same chapter of Matthew which you referred to, and I hope this will satisfy your question, as it was not what they disbelieved about John that is being asked, but instead it is what was not believed about God.