Paul taught thusly, as did the other inspired penman:
Acts 28:23
And when they had appointed him a day, there came many to him into his lodging; to whom he expounded and testified the kingdom of God, -→persuading them concerning Jesus, both out of the law of Moses, and out of the prophets,←- from morning till evening.
The "Q" source wasn't required. Jesus, The Word was already written, and taught in the Law and the Prophets, to which Jesus also expounded, post His Own Resurrection no less:
Luke 24:
25 Then he said unto them, O fools, and slow of heart to -→believe all that the prophets have spoken:←-
26 Ought not Christ to have suffered these things, and to enter into his glory?
27 And -→beginning at Moses and all the prophets, he expounded unto them in all the scriptures the things concerning himself.←-
The New Testament is intimately linked and virtually inseparable from the Old Testament.
Every basic tenet of faith in Christ is found in the O.T. and expounded upon in the N.T. via H.S. inspired engagements, which I might add, is still available today for seeking readers, engaging therein.
John 16:13
Howbeit when he, the Spirit of truth, is come, he will guide you into all truth: for he shall not speak of himself; but whatsoever he shall hear, that shall he speak: and he will shew you things to come.
2 Tim. 3:
14 But continue thou in the things which thou hast learned and hast been assured of, knowing of whom thou hast learned them;
15 And that -→from a child thou hast known the holy scriptures,←- which are able to make thee wise unto salvation through faith which is in Christ Jesus.
They left plenty of scriptural meat on the bone for all of us to reflect upon and engage with.
Nevertheless, Paul was not against any way that Christ was preached, even in pretence:
Philippians 1:18
What then? notwithstanding, every way, whether in pretence, or in truth, Christ is preached; and I therein do rejoice, yea, and will rejoice.
Paul believed Jesus was sufficient to gather and Lead, regardless. Any writings must pass the muster of being built upon the Law, Prophets and Apostles in any cases of sights/teachings/leadings.
Isaiah 8:20
To the law and to the testimony: if they speak not according to this word, it is because there is no light in them.