Mike is a "believer in Q" who is "relying on Stein's book". I like Robert Stein's book but I find it a little unbalanced on the question of Q. Stein mentions the Farrer theory but does not engage with it, and the arguments Stein presents for the existence of Q are a little weak. Mike helpfully summarizes them as follows:
The reason why some argue that Luke is not just copying Matthew is because Luke rarely has Matthean additions to the triple tradition (e.g. Mark 1:32-34/Matt 8:16-17/Luke 4:40-41; Mark 2:23-28/Matt 12:1-8/Luke 6:1-5; Mark 4:10-12/Matt 13:10-15/Luke 8:9-10), places the “Q” material in different contexts (e.g. why would Luke break up Matt’s beautiful Sermon on the Mount?), Matt/Luke never agree in order against Mark, Luke’s lack of “M” material (e.g. the visit of the Magi, Matt’s great commision), “doublets” (sayings appearing in Mark and in “Q” material), etc.I have never been able to find myself persuaded by arguments like these and I attempt some counter arguments in my Case Against Q, and in introductory format in The Synoptic Problem: A Way Through the Maze. In brief, each of these arguments is problematic for a variety of reasons, but I'll pick some of my favourite reasons here and state them as concisely as I can:
"Luke rarely has Matthean additions to the triple tradition": actually, Luke regularly has Matthean additions to the triple tradition but Two-Source theorists re-categorize them as "Mark-Q overlaps" and ignore them for the purpose of this argument.
Luke "places the “Q” material in different contexts (e.g. why would Luke break up Matt’s beautiful Sermon on the Mount?)": Luke treats double tradition sayings material the same way that he treats triple tradition sayings material. Where there are long discourses in Mark (e.g. 4.1-34), Luke retains some, omits some and redistributes the rest, just as when he finds long discourses in Matthew. It is just that there are more of them in Matthew, and they are much longer.
"Matt/Luke never agree in order against Mark": In fact they often do; even Streeter had to build the exceptions into his statement of the supposed rule by noting that they agree all the way through chapters 3 and 4.
Luke’s lack of “M” material (e.g. the visit of the Magi, Matt’s great commision): "M" material is by definition not in Q; of course Luke lacks "M". Luke omits the visit of the magi because of his negative attitude to magi and sorcerers. The Great Commission is reworded at the end of Luke.
Doublets: an argument that would only be convincing if all doublets could be explained source-critically, which they cannot on the Two-Source Theory.
Well, those answers are a little brief, and there is a lot more to be said to tease out the problems with the traditional arguments for Q, but offer these as concise indications that the arguments for Q may need some more thought.