Does Matthew 22:43-45 imply the pre-existence of Jesus?
No, what here would suggest that?
As Jesus did, he quotes a Psalm knowing that they had many of these memorised. He reminds them of what was prophesied and promised, and as was the case many times, it was coming true in part before their unbelieving eyes and minds.
In a similar manner he quotes Psalm 22 on the cross - but only the first and last lines. They knew the rest and all he had to do was get them started and they could join the dots of a prophecy coming true right then about who and what he was.
David is able to, 'in the spirit' (Ps 110), as Jesus reminds, look forward to when his descendant, his Lord to be, would sit at Yahweh's side. He understood, in part at least, what and who was coming.
1Why do the nations rage and the peoples plot in vain? 2The kings of the earth take their stand and the rulers gather together, against the LORD and against His Anointed One (Acts 10:38)... 6“I have installed My King on Zion, upon My holy mountain.” 7I will proclaim the decree spoken to Me by the LORD: “You are My Son; today I have become Your Father. 8Ask Me, and I will make the nations Your inheritance^, the ends of the earth Your possession. Ps 2
Heb 1:2 But in these last days He has spoken to us by His Son, whom He appointed heir of all things
We know when Jesus took his place at God's side. (Acts 2:32-) After his resurrection, ascension, exaltation to the right hand of God with new spirit life. Even Abraham saw Jesus' day and was glad, just as David did.
Your father Abraham rejoiced at the thought of seeing my day; he saw it and was glad. John 8:56
Paul also noted the order of things;
Now the promises were spoken to Abraham and to his seed. It does not say “and to seeds” as of many but “and to your seed” as of One, who is Christ. Gal 3:16
If therefore David calls Him Lord, how is He his son? Matt 22:45
The Jews were aware of the one to come, they knew of this 'other Lord', but they were limited by the veil that God allowed to remain over their understanding.
And even to this day when Moses is read, a veil covers their hearts. 2 Cor 3:15
Can a pre-existent Jesus, who obviously cannot be a man, be David's son? No. We know how he came to be Lord - made so by God and Father Acts 2:36. Clearly David had awareness of God's process and plan and why his inspired Psalms all came perfectly true.
In summary, according to scripture, if Jesus is made heir, appointed, anointed, prophesied to come, foreknown to come, fulfilled prophecy, sits by God's side, made Lord and Christ by God (Acts 2:36) how then could he possibly pre-exist if no verse says he did.
The concept of Jesus pre-existing his conception and birth has no clear revelation in scripture. Only by inference and speculation may this have any credence. By such methodology, many strange ideas have arisen.