Did St. Joachim prevent the transmission of original sin to Mary by overcoming concupiscence? St. Joachim is known to live a pious and humble life. If St. Joachim lived according to God's will and followed His inspirations in making a marital procreative act with St. Anne under the grace of God, is this how original sin was not transmitted to Mary?
The Council of Trent defined the de fide dogma that original sin is transmitted by natural generation. It says:

propagatione, non imitatione transfusum omnibus. (D 790)
In the baptism of children that is expurgated which they have incurred through generation. (D 791)

As original sin is a peccatum naturæ, it is transmitted in the same way as human nature, through the natural act of generation. Although according to its origin, it is a single sin (D 790) that is the sin of the head of the race alone (the sin of Eve is not the cause of original sin) it is multiplied over and over again through natural generation whenever a child of Adam enters existence. In each act of generation human nature is communicated in a condition deprived of grace.
The chief cause (causa efficiens principalis) of original sin is the sin of Adam alone. The instrumental cause (causa efficiens instrumentalis) is the natural act of generation, which gives rise to the connection of the individual human being with the head of the race. The actual concupiscence associated with the act of generation, the sexual pleasure (libido) is, contrary to the view of St. Augustine (De nuptiis et concup. I 23, 25; 24, 27), neither the cause nor the inescapable condition for the reproduction of original sin. It is only an accompanying phenomenon of the act of generation, which in itself alone is the instrumental cause of the transmission of original sin. Cf. S. th. 1 II 82, 4 ad 3.
Based on the Council of Trent definition on concupiscence, it is the &quot;instrumental cause of transmission&quot; and in the Gospel the concupiscence can be overcome as testified by St. Paul (Galatians 5:24-25):

Those who belong to Christ Jesus have crucified the flesh with its passions and desires. Since we live by the Spirit, let us keep in step with the Spirit.

Galatians 2:20:

I have been crucified with Christ and I no longer live, but Christ lives in me. The life I now live in the in the body, I live by faith in the Son of God,who loved me and gave himself for me.

Is there a Church teachings or writings that states St. Joachim may have overcome &quot;concupiscence&quot; and thus prevented the transmission of original sin?
The Blessed Virgin Mary's immaculate conception was not due to the merits of St. Joachim but to "a special grace and privilege of God, in view of the merits of Jesus Christ" (Ineffabilis Deus).

Also, St. Thomas writes (Summa Theologica suppl. q. 41 a. 3 arg. 3) that


  the marriage act is always connected with concupiscenceactus matrimonialis habet concupiscentiam adjunctam (Super Sent., lib. 4 d. 26 q. 1 a. 3 arg. 3)


(Certainly concupiscence in the marriage act can come in degrees, but, as your citation to the Summa mentions, original sin does not.)

Even if the marriage act between Sts. Jochim and Anne involved no concupiscence, original sin would still be transmitted, if it were not for God's "special grace and privilege" to the Blessed Virgin Mary.