Who was the first Church Father to argue for executing heretics? When was it first said in Christianity by a church leader that heretics must be put to death? Who came up with this idea and presented it?
Technically, the first ecclesiastical authority to say that heretics should be persecuted was Pope Gregory IX in 1229.  The first Church Father to say that heretics should be compelled to recant would be Augustine around 400 AD.
The first heretic actually put to death (385AD) was killed by the Emperor - and the Pope was very annoyed at the development. It wasn't for another 800 years that the church followed suit.

That said...
The church typically did not execute heretics. Rather, the state, sometimes at the behest of the church or other times of a monarch desiring no discord amongst his subjects, would carry out such persecution and executions.
The earliest Christian heretics were not put to death by the Church, but rather by the Roman Empire. Note for example, the first &quot;heretic&quot; to be executed - Priscillian - was put to death not by the church, but rather the state.  Furthermore, those responsible for getting the state to put him to death were subsequently excommunicated by the Pope for their actions.
Per this article:

The first person to define what heresy was Irenaeus, Bishop of Lyons (2nd Century).  He simply warned about the dangers of a multitude of opinions on how God works.

The first person to make heresy a crime was Emporer Constantine (320s).  He made the assembling of heretics an offense, the penalty of which was the confiscation of property.

In 380 AD under the Christian Emperor Theodosius I laid down the rule that only the Catholic Christians could define orthodoxy within the confines of the Roman Empire. As noted already, it was under his reign that the first heretic was put to death.

Augustine (AD 354-430) taught that error has no rights. He cited biblical texts, notably Luke 14:16-23, to justify the use of compulsion

Emperor Justinian issued severe laws against heretics in AD 527 and 528. (It should be noted, this was in the aftermath of a severe series of riots that left thousands dead.)

Pope Paschal II (Pope from 1099 and 1118) was the first to say that anyone who disagreed with the apostolic see was a heretic.

Pope Innocent III (1199) was the first to declare heresy to be high treason against God, having already called for the execution of those who persisted in their heresies after being excommunicated.

It was not until 1229 that Pope Gregory IX declared that it is the duty of every Catholic to persecute heretics.



From Justinian:

&quot;Henceforth those who dissented from the authorised line were debarred from public office, forbidden to practice certain professions, prohibited from holding meetings, and denied the civil rights of a Roman Citizen. For them, said Justinian &quot;to exist is sufficient&quot; - for the time being. In the middle of the fifth century Pope Leo the Great commended the Emperor for torturing and executing heretics on behalf of the Church.&quot;


Per Wikipedia on heresy:

The first known usage of the term heresy in a legal context was in AD 380 by the Edict of Thessalonica of Theodosius I,[12] which made Christianity the state church of the Roman Empire. Prior to the issuance of this edict, the church had no state-sponsored support for any particular legal mechanism to counter what it perceived as &quot;heresy&quot;. By this edict the state's authority and that of the Catholic Church became somewhat overlapping. One of the outcomes of this blurring of church and state was the sharing of state powers of legal enforcement with church authorities. This reinforcement of the church's authority gave church leaders the power to, in effect, pronounce the death sentence upon those whom the church considered heretical.
Within five years of the official criminalization of heresy by the Emperor, the first Christian heretic to be prosecuted, Bishop Priscillian, was executed in 385 by Roman officials.
However, his accusers were excommunicated both by Ambrose of Milan and Pope Siricius.[13] For some years after the Reformation, Protestant churches were also known to execute those they considered heretics, including Catholics. The last known heretic executed by sentence of the Roman Catholic Church was Cayetano Ripoll in 1826. The number of people executed as heretics under the authority of the various &quot;ecclesiastical authorities&quot;[note 1] is not known.[note 2]
NOTE 2: ^ Only very fragmentary records have been found of the executions carried out under Christian &quot;heresy laws&quot; during the first millennium. Somewhat more complete records of such executions can be found for the second millennium. To estimate the total number of executions carried out under various Christian &quot;heresy laws&quot; from 385 AD until the last official Roman Catholic &quot;heresy execution&quot; in 1826 AD would require far more complete historical documentation than is currently available. The Roman Catholic Church by no means had a monopoly on the execution of heretics. The charge of heresy was a weapon that could fit many hands. A century and a half after heresy was made a state crime, the Vandals(a heretical Christian Germanic tribe), used the law to prosecute thousands of (orthodox) Catholics with penalties of torture, mutilation, slavery and banishment.[14] The Vandals were overthrown; orthodoxy was restored; &quot;No toleration whatsoever was to be granted to heretics or schismatics.&quot;[15] Heretics were not the only casualties. 4000 Roman soldiers were killed by heretical peasants in one campaign.[16] Some lists of heretics and heresies are available. Thousands were burned at the stake by the Roman Catholic Inquisition, which lasted for nearly seven centuries.[17] From time to time, heretics were burned at the stake by an enraged local populace, in a certain type of &quot;vigilante justice&quot; , without the official participation of the Church or State.[18] Religious Wars slaughtered millions. During these wars, the charge of &quot;heresy&quot; was often leveled by one side against another as a sort of propaganda or rationalization for the undertaking of such wars.