Q. How many scholars believe that Q existed as a source for Matthew and Luke?
A. Many scholars accept the existence of Q as a source for Matthew and Luke. It is a fundamental aspect of the two-source theory, according to which Matthew and Luke independently used both Mark and the hypothetical source, Q. The two-source theory is given as the primary solution to the synoptic problem in most of the introductory textbooks, and it is assumed in many of the commentaries, monographs, and articles written about the Gospels.
However, there have always been detractors, including some who dispute both tenets of the two-source theory, Markan priority, and Q, arguing instead that Matthew was the first Gospel, that Luke used Matthew, and that Mark used Matthew and Luke (The “two-Gospel” or Griesbach theory), and others who maintain Markan priority but dispense with Q by proposing that Luke also knew Matthew (The Farrer theory). It is perhaps safe to say that while Q remains popular among New Testament scholars, there is a significant number of scholars who are unconvinced about the existence of the hypothetical document, Q.
If you’re interested in further reading, I’d recommend these two introductory books: John Kloppenborg’s Q, the Earliest Gospel: An Introduction to the Original Stories and Sayings of Jesus and The Synoptic Problem: A Way Through the Maze, by Mark Goodacre.