Almost certainly not. It's a medieval tale turned conspiracy theory. The legend that there was actually a female pope whose true identity was revealed upon them suddenly giving birth during a procession is fabrication. This would have been the mother of all scandals and, were it true, someone at the time would've mentioned it. The papacy's opponents in eastern churches would have died laughing about it.

The first version of the story was written by Jean de Mailly in his Chronica Universalis Mettensis (Universal Chronicle of Metz) at some point in the mid-thirteenth century. In his version of the story the Popess does not have a name. This is the paragraph of the Chronica that started the legend:

"Pretending to be a man, he was made secretary of the curia by virtue of the righteousness of his character, then cardinal, and finally pope. One day, when he had ridden on a horse, he gave birth to a child and immediately, via the Roman courts, had his feet tied and he was pulled by the horse's tail, and was stoned by the people. And where he died, he was buried, and there it is written: Petre Pater Patrum Papisse Prodito Partum"

I don't know why Jean wrote that, he hasn't based it on anything. The story was picked up in the 1260s by a papal inquisitor name Etienne, who wrote a book of teaching advice called Tractatus de Diversis Materiis Praedicabilibus (Treatise on Diverse Materials for Teaching), where we don't get a name but we do get a date:

"An impressive blow of audacity, or rather insanity, happened around the year 1100, as it is said in the chronicles. A certain educated woman, educated in the art of writing, putting on masculine clothes and presenting herself as a man, came to Rome and, having been accepted, both by her energy and her knowledge, was made secretary of the curia, then with the aid of the devil, was made cardinal and then Pope. She became pregnant and gave birth when she rode a horse and was dragged out of the city. She was stoned by the people... She was buried where she died and... this little verse was written: Parce Pater Patrum Papisse Prodere Partum."

"Around the year 1100" refers to either the pontificate of Pope Urban II, who died in 1099, or of Pope Paschal II, who succeeded Urban. Neither of those popes even slightly align with the story. Certainly, neither gave birth on horseback and got stoned to death in the streets of Rome. However, because this guy was a papal inquisitor writing teaching materials, his work was trusted and disseminated.

Then we get to the 1270s and the Chronicon Pontificum et Imperatorum (Chronicle of Popes and Emperors), by Martin of Opava. He was chaplain to several popes, and his work was viewed as authoritative. Martin places Joan between the pontificates of Leo VI and Benedict III, several hundred years earlier than the previous versions of the story. And, at last, we get a name:

"John Anglicus, born at Mainz, was Pope for two years seven months and four days and died in Rome, after which there was a vacancy in the Papacy of one month. It is claimed that this John was a woman, who as a girl had been led to Athens dressed in the clothes of a man by a certain lover of hers. There she became proficient in a diversity of branches of knowledge, until she had no equal, and, afterward in Rome, she taught the liberal arts and had great masters among her students and audience. A high opinion of her life and learning arose in the city; and she was chosen for Pope. While Pope, however, she became pregnant by her companion. Through ignorance of the exact time when the birth was expected, she was delivered of a child while in procession from St. Peter's to the Lateran, in a lane once named Via Sacra (the sacred way) but now known as the "shunned street" between the Colosseum and St Clement's church. After her death, it is said she was buried in that same place. The Lord Pope always turns aside from the street, and it is believed by many that this is done because of abhorrence of the event. Nor is she placed on the list of the Holy Pontiffs, both because of her female sex and on account of the foulness of the matter."

There is no reliable date for this event, and the two provided (c.1100 and c.855) are totally implausible because there is no gap in the papal records where a pope could be hidden. There are no contemporary accounts of this astonishing alleged event, only manuscripts where scribes in the 1270s have gone back to old manuscripts and written stuff about a female pope in the margin with a little arrow pointing to where it would have gone had it actually happened (which conspiracy hacks point to going "hey look it's historical evidence" without understanding anything about manuscripts). Likewise, all traditions associated with the Joan legend, notably the avoidance of the Via Sacra and chants along the lines of "Parce Pater Patrum Papisse Prodere Partum" all first appear in the later thirteenth century, after the legend had spread thanks to Martin of Opava.

So no, not a historical event. It's a strange tale dreamt up by a chronicler in Metz, spread to teaching material, and then written into the definitive thirteenth century history of the papacy.