There are various passages in the New Testament and early Christian writers which have been interpreted to imply or support infant baptism. However, since you ask for explicit mentions, I'm going focus on ones that are unambiguous. Irenaeaus of Lyon's Against Heresies 2:22:4 hangs on the meaning of "born again" in his era.
The first unambiguous discussion of infant baptism is the argument against the practice in Tertullian of Carthage's On Baptism 18 (198-203 CE). I offer two translations:
Tertullian, as translated by Sydney Thelwall (1885)
according to the circumstances and disposition, and even age, of each
individual, the delay of baptism is preferable; principally, however,
in the case of little children. ... The Lord does indeed say, “Forbid
them not to come unto me.” Let them “come,” then, while they are
growing up; let them “come” while they are learning, while they are
learning whither to come; let them become Christians when they have
become able to know Christ. Why does the innocent period of life
hasten to the “remission of sins?” More caution will be exercised in
worldly matters: so that one who is not trusted with earthly substance
is trusted with divine! Let them know how to “ask” for salvation, that
you may seem (at least) to have given “to him that asketh.” ... If any
understand the weighty import of baptism, they will fear its reception
more than its delay
Tertullian, as translated by Alexander Souter (1919):
in view of the circumstances and will, even the age of each person, a
postponement of Baptism is most advantageous, particularly, however,
in the case of children. ... The Lord indeed says: "Forbid them not to
come unto Me." Let them come, then, while they are growing up; let
them come while they are learning, while they are being taught whither
to come; let them become Christians, when they have been able to know
Christ. Why hurries the age of innocence to the remission of sins?
Shall we act more cautiously in worldly matters? Shall one to whom
earthly substance is not entrusted, be entrusted with heavenly? Let
them know how to seek salvation, that you may be seen "to give to him
that asketh." ... If any should understand the importance of Baptism,
they will be more afraid of its consequences than of its postponement;
unimpaired faith is sure of salvation.
The view that Tertullian was the first unambiguous reference to infant baptism was affirmed by Lutheran scholar Heinrich August Wilhelm Meyer (Commentary on Acts [16:15], New York: Funk & Wagnalls, 1883, p. 312).
Importantly, however, as I read it, the Tertullian passage indicates that others are at least considering infant baptism at his time. Tertullian seems to be responding to someone else quoting Jesus statement, "Forbid them not to come unto Me", in favor of infant baptism.
The challenge of such a late dating of infant baptism is that not long thereafter, Origen of Alexandria, who was incredibly knowledgeable and widely traveled, takes the practice for granted as normative:
Every soul that is born into flesh is soiled by the filth of
wickedness and sin. ... In the Church, baptism is given for the
remission of sins, and, according to the usage of the Church, baptism
is given even to infants. If there were nothing in infants which
required the remission of sins and nothing in them pertinent to
forgiveness, the grace of baptism would seem superfluous
(Homilies on Leviticus 8:3 [248 CE])
The Church received from the apostles the
tradition of giving baptism even to infants. The apostles, to whom
were committed the secrets of the divine sacraments, knew there are in
everyone innate strains of [original] sin, which must be washed away
through water and the Spirit
(Commentaries on Romans 5:9 [248 CE])
It is clear that we are lacking in reliable records on this topic for early centuries.
I should also mention that The Apostolic Tradition, an important work with 3rd century or earlier content, includes this passage:
“Baptize first the children, and if they can speak for themselves let
them do so. Otherwise, let their parents or other relatives speak for
However, the work has an unclear transmission and editorial history, and the majority of the work seems to assume adult converts, so it isn't very helpful in dating the emergence of the practice.