The story you’re probably thinking of is about Reynald of Chatillon, who was captured by Saladin at the Battle of Hattin in 1187. Saladin brought Reynald to his tent:

“He said to him, ‘Here I am having asked for victory through Muhammad, and God has given me victory over you.’ He offered him Islam but he refused. The sultan then drew his scimitar and struck him, severing his arm at the shoulder. Those present finished him off and God speedily sent his soul to Hell-fire.” (Richards, pg. 75)

(In fact I wrote a post about this incident a few months ago - there are several eyewitness accounts but only this one from Baha ad-Din Ibn Shaddad mentions that Saladin offered to let him convert.)

Most crusaders probably wouldn’t have taken this offer, especially if it was ever offered to members of military orders like the Hospitallers or Templars. In their case it would have only been to attempt to humiliate them or tease them before executing them, since they were considered to be particularly vicious enemies of Islam and they were executed instead of being taken prisoner.

But most knights could be taken captive and it seems that sometimes they did convert. We don’t really have any accounts written by them personally, but occasionally they are mentioned in other sources. For example, the Damascene poet and ambassador Usama Ibn Munqidh often saw captive crusaders around Shaizar, his family’s castle in northern Syria. They weren’t always knights, but sometimes included women and children as well:

“Among those Frankish captives who passed into my father’s household was an old woman accompanied by her daughter - a young woman, beautifully formed - and a son who had come of age. The son converted to Islam and was quite a good Muslim, judging from what one saw of his praying and fasting. He learned the craft of working with marble from a stonecutter who used to cut stone for my father’s house. After staying for a long time in my father’s household, my father gave him as wife a woman from a pious family and paid all the expenses required for his wedding and home. His wife bore him two sons and they grew well and their father was pleased with them. When the boys were six or seven years old, their father, the attendant Ra’ul, took them and their mother and everything he had in his house and, the next day, joined the Franks at Apamea. There, he and his sons became Christian again after having been Muslim, despite all their praying and fasting.” (Cobb, pg. 143)

So here, a woman and her children were taken captive in a raid, and the woman’s son converted to Islam. Usama renders his name as “Ra’ul” so his name was probably Raoul in French. Some time later, he had his own family, and eventually they fled back to the crusaders in nearby Apamea. But this was many years later - probably at least ten years or more. This is a recurring theme whenever either side wanted to convert. Could they be trusted? Usama suggests that Christians who became Muslims were faithless and would run away as soon as they could. The crusaders were paranoid about this as well - and so were Christians in Europe. In Spain for example they were always afraid that “conversos” were actually “crypto-Muslims” (or “crypto-Jews”), who pretended to be Christian but still practised their old religion.

On the crusader side, they were also worried that people would just simply go over to the Muslim side willingly, without being captured in battle or in a raid. In the 1160s, the historian William of Tyre wrote about this to Pope Alexander III, who responded:

“You have related to us that it often happens in your province that one of those who are tied to each other in the bounds of marriage either willingly goes over to the enemies of the Christian name and faith, or is sometimes carried there unwillingly. While they are there they also renounce their faith and put away the Christian religion, and they marry another, and they do not return to their spouse who remains an inhabitant of their former house, but entirely neglect to go back. The ones who remain at home, wishing to observe the integrity of the faith, and never admitting the companionship of another person, after a long period of waiting, fatigue you with their questioning and on many occasions ask that you grant them the ability to take a second vow. In the same way you suffer grave troubles concerning these people, who have either been captured by the infidels, or carry themselves over to the superstition of the enemy by denying the faith of Christ, while in either case the other spouse remains a pure believer in the Trinity, until divine grace leads the other spouse to return. But because you seek our response concerning these matters, we respond to your request that, since they are made one flesh in the union of marriage, matrimony cannot be dissolved for any reason without the judgment of the church, nor may license to take a second vow be granted to them at any time, because as the holy scripture witnesses, a man cannot dismiss his wife except because of fornication, and then he must either be reconciled to her or remain celibate while she is still alive.”

In short, if one spouse runs away and becomes a Muslim, the other spouse can’t remarry because they are still married according to the church. The church held out hope that the spouse who converted might change their mind and come back.

The crusaders themselves did not seem to have the same hope. Around 1200 - so, the decade after Hattin and the Third Crusade, when the crusader kingdom was re-established along the coast, with it’s capital at Acre - one crusader legal text says that if an unmarried knight flees to Muslim territory, his property will be confiscated by his lord. But if the knight is married, his wife can inherit his territory as long she marries someone else. This is the opposite of what Pope Alexander suggested. The same set of laws say that if a knight is captured in battle and is returned, he can keep his land, as long as he swears that he never converted during his captivity. So they must have considered that conversion was a likely outcome and didn’t quite fully trust anyone who came back.

Later in the 13th century it seems that anyone who ran away (or had been captured) and converted, and then came back, was pretty much ostracized and left to their own devices. Pope Urban IV, who had been patriarch of Jerusalem before becoming Pope and probably understood the situation well, wrote a letter to the crusaders in 1264 about two specific converts:

“Since therefore our beloved sons the lay brothers Peter and Andrew of Bethlehem, who have brought us the present letter, have, as they assert, returned to the practice of Catholic purity from the filth of the pagans, with the Son of God supporting them, and since they do not have the means to sustain themselves if they are not supported by the subsidy of others, we ask you, our brother, and urge you attentively, ordering you through our apostolic writings that if you agree that the aforementioned laymen have been baptized or have returned to the faith, and there has been no other provision made for them, you should have the necessities of life provided for those same men in any monastery or religious site in the patriarchate of Jerusalem which is not already burdened with a similar provision…”

So we don’t have any accounts by the converts themselves, but other sources do take note of them, and it seems that people would convert willingly or unwillingly often enough that it became an issue for the church and for the crusader legal system. They were supposed to be supported, like the “prodigal son” in the Gospels, but in reality they were looked at with suspicion and distrust by both Muslims and Christians.

Sources:

Baha ad-Din ibn Shaddad, The Rare and Excellent History of Saladin, trans. D.S. Richards (Ashgate, 2002)

Usama ibn Munqidh, The Book of Contemplation: Islam and the Crusades, trans. Paul M. Cobb (Penguin Classics, 2008)

Yvonne Friedman, Encounter Between Enemies: Captivity and Ransom in the Latin Kingdom of Jerusalem (Brill, 2002).

Benjamin Z. Kedar, "Multidirectional conversion in the Frankish Levant", in Franks, Muslims, and Oriental Christians in the Latin Levant (Ashgate, 2006)

Benjamin Z. Kedar, Crusade and Mission: European Approaches Toward the Muslims (Princeton University Press, 1988)