Tertullian: Earliest extant use of term
Tertullian (155 - 240) was, famously, the first Latin writer to use the word "trinity" in his anti-Sabellian treatise Against Praxeas. It also contains the earliest known uses (16 total) of the word "persons" in reference to the Godhead.
The mystery of the dispensation is still guarded, which distributes the Unity into a Trinity, placing in their order the three Persons— the Father, the Son, and the Holy Ghost: three, however, not in condition, but in degree; not in substance, but in form; not in power, but in aspect; yet of one substance, and of one condition, and of one power, inasmuch as He is one God.
Valentinus: Alleged earlier use of term
Marcellus of Ancyra (d. 374) claimed in his own treatise On the Holy Church (paywall) that Valentinus the Gnostic (100 - 160) was the first to propose three persons in the Godhead:
These then teach three hypostases, just as Valentinus the heresiarch first invented in the book entitled by him On the Three Natures. For he was the first to invent three hypostases and three persons of the Father, Son and Holy Spirit, and he is discovered to have filched this from Hermes and Plato.
There are significant reasons to doubt Marcellus' assertion:
On the Three Natures has not survived in any manuscripts or fragments, and we have no way of knowing whether Valentinus used the words "hypostases" or "persons" or if one or both was an interpretive flourish by Marcellus.
If it's specifically the word "hypostases" that Marcellus objects to, then it would seem to be an anachronistic reading of On the Three Natures, which was written centuries before the word was first used as a synonym of "persons."
Marcellus wrote this specifically to discredit the orthodoxy of the term "persons" by linking it back to a despised heretic, which lends credence to the "interpretive flourish" hypothesis.
It is also exceedingly unlikely that Tertullian borrowed the terminology from Valentinus, since he himself was a fierce critic of Valentinianism, even dedicating a whole book to it, Against the Valentinians.
We have no surviving writings from anyone prior to Tertullian using the word "persons" in a trinitarian fashion.
Whether Tertullian was or wasn't the very first to use the term cannot be proved. He has a reputation as an innovative theologian, but whether he was innovative in service of defense of orthodoxy or in service of inventing new heresies is a matter left to the reader. Of course, even if the concept was orthodox, that doesn't automatically mean that his use of the word had any precedent. In the absence of other evidence, we have no cause to say that anyone else was the originator of the use of the word.