The debate is raging about the nature, origin, and even existence of Q. I think a major problem is: How could the author(s) of Q not embrace the resurrection of Jesus? What is key here to me is this: Would the author(s) of Q really see Jesus' sayings as significant and even authoritative if they knew that he had died as a criminal? It is hard to believe that they would venerate his teachings and not know about his death. But if that is the case, how could they then promote Jesus--unless they also embraced the resurrection? It raises an interesting problem for diverse Christianities in the earliest period. Further, scholars usually recognize that Q is a Galilean source. Some suggest that it was composed shortly after Jesus' ministry in Galilee, and began circulating perhaps even before his death. Hence, the reason there is no mention of his death and resurrection as a fact (this is the view that J. D. G. Dunn recently suggested in his Jesus Remembered).
My own take on Q is that the authors saw Jesus as a sage. What was most important to them were his sayings, as is the case with other wise men. These sayings then were compiled and circulated in Galilee. When, exactly, the circulation of them would have taken place is difficult to know but almost certainly no later than the 40s. If that late, the authors of course would know the story of how Jesus' life ended, and would also know about the vindication of Jesus by his resurrection. That would at least put a stamp of approval on Jesus' sayings! It would certainly do more than that, but Q was concerned with the logia, not the life. The reason for this can be variously explained. Dunn suggests that it is fallacious to think of the Q community as playing only one song; the Dead Sea Scrolls have demonstrated that to us! Q would have been used in various Christian communities, but so would Mark and other materials. One document doesn't tell us what a community thought en toto about Jesus. (James seems to function in a similar way. I think it's the earliest NT book, written in the mid-40s, and focusing much more on the words of Jesus than his life. It alludes to the sermon on the mount several times and only mentions Jesus twice.) At the same time, I do think that the early apostolic community was searching for a handle on how to think about Jesus. Certainly, the resurrection proved to them that God had blessed him. And it proved to them that he was a prophet and the Messiah. But how much more they would have understood is difficult to know. Even Peter's Pentecost sermon does not speak of believing IN Jesus: one believes in God ABOUT Jesus. While the early Christian community was sorting out what Jesus meant, they at least knew that they could trust his words.
Hence, I believe that Q never could have gotten off the ground, never could have become a source of information for later gospel writers, unless Jesus had been raised from the dead. It circulated sufficiently that BOTH Matthew and Luke had access to copies of it.