I decided to post something on this topic because someone recently sent me a post promoting (?) a particular view of “Q” and Matthean authorship that appeared at divinecouncil.org. That is not my website.
Briefly, the website was promoting the work of New Testament scholar Alan Garrow. Back in 2004 Garrow published a study (his Oxford dissertation) that argued that the writer of the Gospel of Matthew was dependent on Q for his gospel. For those who don’t know, Q is the abbreviation (from German quelle, “source”) for the presumed source of the gospels Matthew and Luke. The argument is made in part because Matthew and Luke share a lot of material that does not appear in the third synoptic gospel (Mark). John isn’t one of the synoptics because 90% of John is unique to John. Scholars explain this sharing by arguing that Matthew and Luke must have both used the same source — a source now missing. Hence the hypothetical source is called Q. The Didache is an ancient Christian document dated by many scholars to the first century. It is essentially a catechism and manual of ethics. Garrow’s work argues that the Didache is Q — the lost source of Matthew and Luke.
I’m not a New Testament specialist, so by definition I’m not an expert on the synoptic problem. Readers should know that Garrow’s work is regarded as important, but its reviewers remain unconvinced. The book is highly technical, but here are some journal reviews of it for those interested. The ones by Kelhoffer and Kloppenborg are (to my eye) the most helpful.
Lastly, not all scholars believe there was a Q. Mark Goodacre at Duke is perhaps the leading mainstream New Testament scholar skeptical of Q. You can read his thoughts on his website devoted to the topic: The Case Against Q.