Ronald A. N. Kydd in his book Charismatic Gifts in the Early Church mentions several second-century quotes that he argues refer to glossolalia, what we call speaking in tongues. His work clearly deals with the first century as well, but in that period he mentions only prophecy, not tongues.
The first evidence for speaking in tongues that Kydd mentions comes from the Ecclesiastical History (5.16) of Eusebius. He cites the second-century history of Apolinarius of Hierapolis, who criticizes Montanism. Describing the movement's namesake, the text reads:
And he became beside himself, and being suddenly in a sort of frenzy and ecstasy, he raved, and began to babble and utter strange things, prophesying in a manner contrary to the constant custom of the Church handed down by tradition from the beginning. (§7)
And he stirred up besides two women, and filled them with the false spirit, so that they talked wildly and unreasonably and strangely (§9)
Kydd argues that this is glossolalia, though the text does not state this explicitly. Kydd makes the same argument about an ambiguous passage in Origen's writings Against Celsus, 7.9. Celsus wrote his work around AD 177, criticizing Christianity, and Origen describes a portion of it as follows:
Then [Celsus] goes on to say: “To these promises are added strange, fanatical, and quite unintelligible words, of which no rational person can find the meaning: for so dark are they, as to have no meaning at all; but they give occasion to every fool or impostor to apply them to suit his own purposes.”
I'll briefly mention that others reject these understandings, such as Nathan Busenitz, arguing that these references are not actually to glossolalia.
The clearest reference to glossolalia from the second century, then, comes from Irenaeus, Against Heresies, 5.6.1, written around AD 180:
In like manner we do also hear many brethren in the Church, who possess prophetic gifts, and who through the Spirit speak all kinds of languages, and bring to light for the general benefit the hidden things of men, and declare the mysteries of God, whom also the apostle terms “spiritual,” they being spiritual because they partake of the Spirit, and not because their flesh has been stripped off and taken away, and because they have become purely spiritual. [emphasis added]
This quotation is more clearly understood as glossolalia, though there is debate over the nature of that – for more on the question of what these tongues were understood to be, see: Do any church fathers directly connect “speaking in tongues” with anything other than existing human languages?