You are right that William Marshal, 1st Earl of Pembroke, was an extremely important figure in English - and even in world - history. It can be argued that, as protector and regent for King Henry III, he was responsible for the re-issue of Magna Carta in 1217. Indeed, he is often referred to as "The Greatest Knight in Christendom", although - to be fair - there are other rivals for that title!
He did, indeed, travel to the Holy Land, and you are right to say that this episode of his life is particularly poorly recorded. In fact,everything that we know about the episode is recorded in his 13th century biography, L'Histoire de Guillaume le Marechal. This was originally written in medieval French, but a translation into modern French is available on Archive.org. (I did find some English translations online, but the copyright status of these volumes appears questionable at best!). You should be able to find a copy of Nigel Bryant's English translation in a decent library near you.
What We Know:
We know that William Marshal accepted the bequest of the Crusader Cross of Henry the Young King, the son of Henry II, when Henry was on his deathbed. We know that Henry died on 11 June 1183 which gives us one fixed date.
William Marshal's biography states that he spent two years in the Holy Land. We are told that he served the King of Jerusalem, and that he performed great deeds of arms. We're told that he earned the respect of both the Knights Templar and Knights Hospitaller He didn't join either order, but fought as a lay brother with the Templars. He did however take a vow to join the Knights Templar before he died.
He seems to have been back in Europe by 1187. We're told that he brought two shrouds of white Damascus silk for his own burial. He did join the Templars before his death in 1219, and was buried in the Temple Church in London (you can still see his tomb there).
And that's about it!
Now, we know a lot about what was happening in the Holy Land at this time, even if we don't know exactly what William Marshal's role was in those events. You'll find the details in any good history of the Crusades, for example, the second volume of Stephen Runciman's History of the Crusades. However, there are a couple of points that may be of particular significance.
Guy de Lusignan became King of Jerusalem in 1186. Now, we know that William Marshal had previously had a run in with the Lusignans when he was younger.
One of Guy's principle supporters in Jerusalem was the Grand Master of the Templars, Gerard de Ridefort.
One other detail from William Marshal's biography may be relevant here.
In 1168, William Marshal had been part of the entourage escorting Queen Eleanor through her own territories near Poitiers when they were ambushed by the Lusignans. In the course of the ambush, Patrick of Salisbury was killed and William Marshal himself was taken prisoner. Patrick of Salisbury was unarmed and not wearing armour when he was killed, which was a breach of the Chivalric code (and the reason why the episode is often referred to as the murder of Patrick of Salisbury).
We're also told that William Marshal was mistreated by the Lusignans while he was their prisoner. However, as far as I'm aware - and not withstanding what it says in the Wikipedia article - we cannot be certain which of the four Lusignan bothers were involved.
What we can infer:
Given William Marshal's martial prowess, it seems likely that he would have been in the forefront of the military campaigns in the Holy Land at this time. His reticence on the subject is surprising, although it may be explained by the fact that he had left Outremer and returned to England just months before the disastrous (from the perspective of the Crusader Kingdom) Battle of the Horns of Hattin in July 1187. From what we know about William Marshal, it may well be that he felt that he had 'abandoned' the men he had fought with, and considered that a stain on his honour.
We know that William Marshal valued honour just about above all else, and from the chronicles it would seem that he tried to live his life strictly according to the laws of Chivalry. It then seems reasonable to infer that, if Guy de Lusignan had been involved in the murder of Patrick of Salisbury (and the mistreatment of William himself while he was their prisoner), then he might have been reluctant to serve under a man who he felt lacked honour.
Given that antipathy, he might equally have felt that he could not join the Templats while they were commanded by a man who supported Guy de Lusignan.
This would also explain why he left the Holy Land when he did.
However, all this remains speculation - even if it is based on the available evidence (although I have seen it stated as 'fact' in some books about William Marshal).
If it is correct though, then it is ironic that in later life, William Marshal would find himself bound by oath to serve King John, probably the worst king in England's history, and a man who was notoriously lacking in honour!