No need for a Q source theory when the hermeneutics of sensus plenior are understood. The differences in the gospel result from study of the OT in light of Christ. The hermeneutic can be reverse engineered out of the differences to show what 1st century interpretative tools were used.
The source is the Old Testament as interpreted using the SP hermeneutic.
The Q theory requires that you subscribe to the implausible idea that the apostles did not communicate with each other over the course of their various ministries.
The first church needed no New Testament.. they didn't have one yet taught the scriptures they had. The Greek church needed an interpretation of the Hebrew teaching. The gospels were produced for them.
Each is a snapshot of the current state of Bible teaching over time as the apostles remembered what Jesus taught. Each book reflects a greater proficiency in handling the tools of sensus plenior interpretation. Paul called it "the mystery, hidden from the beginning".
The Q theory is nothing but speculation. The SP hermeneutic is available for all to examine and validate. Study to show yourself approved.