There wasn't "an" event, but rather the circumstances that the revolutionaries found themselves in influenced Robespierre's view on capital punishment.

In order to understand Robespierre's stance on using death as a punishment it's important to understand the idea that there were two different "stages" (so-to-speak) of government: revolutionary government, and post-revolutionary government (a standard, permanent government). As outlined in one of Robespierre's most famous speeches ("On Morality"), he saw the revolutionary government as existing outside of the normal bounds of rules and practices that would constitute the ideal government during peace time. The revolutionary government necessarily had to utilize punishments such as the death penalty, Robespierre reasoned, due to the fact that those fighting against them, who would bring the monarchy back and re-yolk the people to an unjust regime, would use violence without flinching. In "On Morality", Robespierre lays out his position on fighting violence with violence, as it were. He argues that to let the virtuous, good people be killed and opressed by those who were bad was not justice ("is not the lightning of Heaven meant to strike vice exhaulted"). Perhaps his most quoted line is a good illustration on his view of violence at this time. I'll put it here in it's context in "On Morality":

If the mainspring of popular government in peacetime is virtue, amid revolution it is at the same time [both] virtue and terror: virtue, without which terror is fatal; terror, without which virtue is impotent. Terror is nothing but prompt, severe, inflexible justice; it is therefore an emanation of virtue. It is less a special principle than a consequence of the general principle of democracy applied to our country's most pressing needs.

In this anyone who has read about Robespierre can hear his voice perfectly. This was Robespierre at his peak (1793-1794). He didn't believe that he was contradicting his earlier stance against the death penalty. The capital punishment (and especially gruesome torture) of the ancien regime was used frequently, and often without due justice given to the accused before they were executed. That regime was not 'worthy' of wielding the awesome responsibility of capital punishment-- only the virtuous were. Now Robespierre did not foresee the death penalty being necessary in the Republic of Virtue, either. This government that would emerge forged in the fires of Revolution would have a virtuous citizenry-- the unvirtuous would have been executed during the Terror, or fled abroad-- and therefore would have no need to resort to capital punishment. As Robespierre (and the many other acolytes of Rousseau) believed, goodness was innate, written "not in marble and stone, but in the hearts of all men". If anyone were to go astray under a Republic of Virtue, physical force would be entirely unnecessary, as reason, logic, and community could set them straight in their thinking (this is, for example, how anouther Rousseau-ist Louis Mercier viewed the future of crime in his pre-revolutionary best-seller L'An 2440).

So to go to the heart of your original question, of whether a single event (or events) influenced Robespierre, I would say that the circumstances of the times developed his view, not that his view necessarily changed as dramatically as it's portrayed by those displaying a less nuanced understanding of the Revolution in general, and Robespierre specifically. There is a famous theory of how the Terror came to be that fell out of favor in the late 19th century but is beginning to come back into favor: Aulard's "Theory of Circumstances". This Theory seeks to explain the seemingly 'inexplicable' (at least to certain historians) violent turn of the Revolution; from idealistic political revolution in 1789, to violent social revolution in 1793. I personally find the Theory of Circumstances very compelling, because its central hypothesis is that we cannot remove the humanity of those involved from the Revolution. To be more clear, we cannot sit in comfortable chairs and shake our heads at the 'horror' of the Revolution, cannot denounce Robespierre and the other Revolutionaries as monsters, without seeking to understand the feelings and the emotions and the sheer panic and chaos that embodied the reality of the Revolution. So often, especially it seems in the Revisionist versions of the French Revolution, the "reality" seems to be stripped away-- the sans-culottes become an evil, mean roaming band, set on violence and revenge, terrorizing the 'upstanding' classes who merely were attempting to advocate for their liberties. Outside of being a very elitist view of history, it disregards the question of why they felt such acts to be necessary. These events were not happening in a vacuum, and the terror of the foreign war, the abject fear at a resurgent aristocracy returning to power and slaughtering the patriotic, the scarcity of basic food and supplies, all of these things were very real, and happening to people just as alive as you and I. Their emotions, their doubts, their uncertainties colored their decisions at every step of the way.

In this Robespierre was no different. As the Revolution wore on there were no guarentees that the foreign armies wouldn't crush the army and pour into Paris. There was no guarentee that a counter-revolution wouldn't rise up and massacre the revolutionaries, re-entrenching Bourbon-ministerial absolutism. As events twisted and turned-- one moment towards Constitutional Monarchy, the next towards Republicanism-- the revolutionaries postured and adapted. Every single man, woman, and child was experiencing an event heretofore unknown to them or their families, and that experience of the unknown was terrifying and exhilarating (as the diaries of various Revolutionaries makes clear). No one event caused a shift in thinking for Robespierre, rather he molded his beliefs to fit the situation facing France. It's illustrative to imagine a single ball of clay that were Robespierre's principles that he could mold and shape to fit the situation, rather than imagining him tossing out his old clay ball and grabbing a new one as events changed.

Hopefully that helped answer your question! Let me know if you have any follow-ups!