What is first time infant baptism was established and commissioned according to catholic church? What does church history say about this?

What [does] church history say about this ?

Since two wrongs make a right, here are two technically non-Catholic, but ultimately Catholic-friendly snippets; to make a long story short, it's ultimately due to Christianity's historically Jewish roots:

A significant parallel exists between Jewish proselyte baptism (when pagans were converted to Judaism) and early Christian baptism. The contacts between early Christian baptism and proselyte baptism, with the similarities in terminology, interpretation, symbolism, and the rite itself, are especially notable. What is of greatest interest, however, is that the baptism of the early Church followed that of proselyte baptism, in which children and infants were baptized with the convert’s family. This is especially significant when one realizes that the very early Church was made up primarily of converted Jews.

— Jordan Bajis, Infant Baptism.

The New Testament church is not a new and different church, but one with that of the Old Testament. The terms of admission into the church have always been the same viz., a profession of faith and a promise of subjection to the laws of the kingdom.

Now it is a fact beyond dispute that the children of God's people under the old dispensation were recognized as members of the church. Circumcision was the sign and seal of their membership. It was not because of carnal descent from Abraham, but as being the children of God's professing people, that this rite was administered (Romans 4:11). If children were members of the church under the old dispensation, which they undoubtedly were, then they are members of the church now by the same right, unless it can be shown that they have been expressly excluded.

Under the Old Testament parents acted for their children and represented them. (See Genesis 9:9; 17:10; Exodus 24:7-8; Deuteronomy 29:9-13). When parents entered into covenant with God, they brought their children with them. This was a law in the Hebrew Church. When a proselyte was received into membership, he could not enter without bringing his children with him.

The New Testament does not exclude the children of believers from the church. It does not deprive them of any privilege they enjoyed under the Old Testament. There is no command or statement of any kind, that can be interpreted as giving any countenance to such an idea, anywhere to be found in the New Testament. The church membership of infants has never been set aside. The ancient practice, originally appointed by God himself, must remain a law of his kingdom till repealed by the same divine authority. There are lambs in the fold of the Good Shepherd (John 21:15; Compare Luke 1:15; Matthew 19:14; 1 Corinthians 7:14).

— Easton's Bible Dictionary: Baptism, Christian.

What is first time infant baptism was established and commissioned according to catholic church? What does church history say about this?

According to Catholicism infant baptism was initially started by the Apostles themselves.

The reason for infant baptism being established was not the decline of membership of the Early Church! History shows that membership in the Early Church increased steadily by leaps and bounds. The pagan converts quite soon in the Early Church outnumber Jewish converts to Christianity.

The baptism of infants has always been maintained in the Catholic Church because the Church believes that it was initially done by the Apostles themselves; baptism removes original sin from the souls of infants; and having been baptized as an infant makes eternal life possible for children who die during infancy.

Does baptism of infants have any biblical foundation

Right from its inception, the Church has been administering the sacrament of baptism not only to adults, but to infants and children as well. In his first powerful and his inspired preaching, after the Pentecost experience in Acts 2: 1-13, peter invites his audience to repent and be baptized, so that they may be saved. “For the promise of God was made to you and your children” (Acts 2:39) Children are therefore invited to receive salvation and since baptism is the necessary means of receiving salvation, children cannot and should not be denied the sacrament of baptism.

Lydia, a God-fearing woman from Thyatira City, was baptized together with her household (Acts 16:15); and Paul and Silas baptized the former guard of her jail and his whole household (Acts16:35). A leading man from Crispus, along with his whole household, became believers and were baptized (Acts ); Cornelius and his househould became God-fearing and received baptism (Acts 10:2); and Paul baptized Stephana’s family (1 Corinthians 1:16). Though one cannot conclude with certainty that infants and children were baptized in this synopsis of biblical passages above, the term “household” presupposes that infants and children were included in the reception of the sacrament of baptism. Moreover, one cannot fathom how Lydia, Cornelius and the prison guard could possibly exclude their children from receiving salvation, which is made through baptism after accepting Jesus Christ as their Lord and Savior.

It is on the basis of the aforementioned understanding and context that the Catholic Church administers the sacrament of baptism to infants and children even though they lack the capacity to make a personal profession of faith in the Lord Jesus. Another traditional reason for this practice is that baptism washes away original sin. Infants and children are baptized on the condition that their parents will bring them up in the practice on the faith.

One could ask the reason, why did Peter order baptism of Cornelius’ household even though they had received the holy spirit?

The answer is as follows:

The story of Cornelius and his family stands out because it is the only time that someone is recorded as receiving the Holy Spirit before baptism.

However it is important to note that the Church does not teach that God is bound by the sacraments (Catechism, 1257). God can impart his grace and the Holy Spirit upon whomever he chooses, whenever he chooses.

The fact that Peter chose to baptize Cornelius and his family is evidence of how serious the Apostles considered the reception of the sacraments to be. It would appear to me that Peter is performing the equivalent of the modern concept of a “conditional baptism.” There are times when it appears that an individual might very well have been validly baptized but there’s no definitive proof, at such times the priest would perform a conditional baptism to ensure the sacrament was properly received. Peter, it appears to me, is doing something similar. It is apparent to him that Cornelius has received the Holy Spirit but has not received the sacraments in an ordinary manner. Peter was “covering all the bases” and avoiding having to judge future claims of individuals having received the grace of God directly from God. - Why Baptize Cornelius If He Already Received the Holy Spirit?

Infant baptism was not established because of the decline of membership of the Early Church. The exact opposite is historical known.

Most scholars agree that once Christianity appeared, it grew at a compounded rate of approximately 3.4 percent per year, becoming the religion of about half the empire by 350 AD, eventually reaching a majority. Since the empire already had an existing religious system, the central questions for historians of the last 250 years have been "how – and why – did this happen?". While the growth of Christianity in its first three centuries "cannot be simply explained by the decline of paganism", it is the fourth century that has attracted the most debate. Until the end of the twentieth century, the primary theory for what caused the dramatic cultural and religious change of the Roman empire revolved around Constantine the Great (r. 306 to 337). - Christianization of the Roman Empire

Not sure if this is the first recorded instance (Catholics would contend that the first recorded instance is in Acts 10 ), but it appears it was a matter of contention in the 3rd century:

But in respect of the case of the infants, which you say ought not to be baptized within the second or third day after their birth, and that the law of ancient circumcision should be regarded, so that you think that one who is just born should not be baptized and sanctified within the eighth day, we all thought very differently in our council.

St. Cyprian of Carthage - Epistle 58

So weren't as concerned about adult vs child Baptism as 3 day old vs 8 day old Baptism. Either way, it's infant Baptism.

Also, if you parse what St. Cyprian is saying, Baptism mirrors Circumcision for Christians so it wouldn't make a ton of sense to postpone Baptism to adulthood.