There isn’t any; and the Church neither supports nor rejects ‘Q’.Please provide references and official church documentation for this.
However, there are other ways of reconciling the Synoptic Problem without introducing ‘Q’. The main problem with Q is that it’s completely hypothetical, and hasn’t been found even in fragmentary forms. The Q theory also ignores the witness of the early Church writers, who attest to an existing Matthew in Aramaic (e.g. Papias). Even if there was a chance that they were in error, it is a mistake to disregard them.
I tend to lean towards the solution proposed by the 1966 Jerusalem Bible, which proposes an Aramaic Matthean priority (consistent with early Patristic testimony), Markan Greek priority, and with Greek Matthew and Luke referencing Mark and Aramaic Matthew (IIRC, I’ll have to look it up).
Unfortunately, some would rather resort to polemics rather than reason when presenting their objections to Q or other modern hypotheses.