Definitely! There were some in the 13th century for sure, but jousting was probably introduced to the east with the crusaders in the 12th century as well.
Probably the most famous one took place in 1225, during the knighting ceremony for the two eldest sons of John of Ibelin, the “Old Lord of Beirut”:
“It happened that my lord of Beirut made his two eldest sons knights in Cyprus. The one was Sir Balian, who later became constable of Cyprus and lord of Beirut; the other was Sir Baldwin, who became seneschal of Cyprus. At this knighting was the greatest celebration and the longest which was ever held this side of the sea to anyone’s knowledge. And there was a deal of giving and spending and jousting, and there were reproduced the adventures of Brittany and of the Table Round, and there were many manner of games.” (Philip of Novara, pg. 66)
Philip of Novara implies that tournaments like this happened frequently. This one is especially interesting because they were basically cosplaying as Knights of the Round Table. Dressing up and pretending to be King Arthur and his knights was a popular thing to do in England and France in the 14th century, so it’s fun to see that knights were doing that in the 13th century as well, all the way on the other side of thee Christian world.
Tournaments were supposed to be a sport so participants weren’t supposed to have hard feelings about anything that happened between the lists (“the lists” are the boundaries of the jousting area). But during this tournament in 1225, a knight named Amaury Barlais was accidentally wounded by another unnamed knight. The next day, Amaury attacked and wounded the other knight. Amaury had to flee to the mainland to escape punishment.
Another notable tournament took place on August 15, 1286, when King Henry of Cyprus was also crowned king of Jerusalem. This was a pretty much meaningless title in 1286 since the crusaders had lost Jerusalem long before, but:
“The coronation was followed by lavish celebrations with jousts and other entertainments - 'the best that had been seen for a hundred years’.” (Edbury, pg. 97)
Apparently this tournament was even bigger than the one in 1225. They didn’t have much to celebrate after that though. Henry was still king a few years later in 1291, when all the mainland crusader territories were lost.
Unfortunately I don’t think we have any records of tournaments like this in the 12th century, but we know they must have happened because they spread into the Byzantine Empire as well, either from the crusader states or maybe directly from Europe (or from both directions):
“[Emperor Manuel I Komnenos] took a particular interest in the customs of western Europe. He was prepared to participate in tournaments, a Western institution, hitherto unknown in Byzantium.” (Harris, pg. 110)
There is even a description of one of these jousting tournaments - the description is from the 14th century, but it's probably referring to Manuel I:
“[Here are] heroic riders on a plain. [Here is] Ares unstained by blood. Graces run around [the scene]. They hold up crowns and crown the heroes. Some of [the heroes] swagger for the most part and are borne high on noble horses. As for their horses, not one of them resembles another. One is white, another black, another blood-red bay, another chestnut, and another speckled.” (Jones and Maguire, pg. 105)
The author makes a lot of references to Homer, so it’s possible the language is entirely figurative, but maybe the participants dressed up as mythological heroes? Just like the ones who dressed up as Arthurian nights on Cyprus.
When the crusaders conquered the Byzantine Empire during the Fourth Crusade, they introduced jousting tournaments to their new territories there as well:
“The prince [of Achaea] sent his messengers throughout all of Romania [the Latin Empire] and all the islands [in the Aegean], ordering them to announce that seven pilgrims who had come from the Holy Land were challenging all knights who wanted to come joust and ‘lose a horse or win a horse’. The jousts would last twenty days and would be held at the city of Corinth…Then Prince Philip of Savoy came and jousted nobly, as did all the knights of his house.” (Chronicle of Morea, par. 1016-1017)
Then the Duke of Athens showed up and, just like on Cyprus, there was some shenanigans, some post-jousting fighting and violence, and what happened in the lists did not stay in the lists!
Jousting tournaments as entertainment probably developed out of the concept of judicial duels, which were originally meant to settle disputes in an organized way, so the parties still got to fight without resorting to vigilante violence. Judicial duels were probably held in the crusader states from the very beginning. For example, as early as 1124, Count Hugh II of Jaffa was accused of treason against the king. He denied it, and was challenged to a duel by one of the king’s men. (In that case, he didn’t show up to the duel, and he was convicted in absentia.)
Another well-known judicial duel took place in 1228. Amaury Barlais, the same guy who attacked a knight after the tournament in 1225, also challenged another knight to a duel a few years later. The other knight, Anceau of Brie, injured Amaury during the duel. According to the rules of the duel, Anceau would have been allowed to kill Amaury, but John of Ibelin stepped in and saved Amaury’s life. Since Amaury had accused Anceau of a crime, but then lost the duel, he had to pay a humiliating fine.
So, in brief, yes there were definitely jousts and tournaments in the crusader states in the Near East, on Cyprus, and in former Byzantine territories. The crusaders even seem to have spread tournaments to the Byzantines. There were also judicial duels or combats, which were a more serious kind of jousting.
Sources:
Peter W. Edbury, The Kingdom of Cyprus and the Crusades, 1191-1374 (Cambridge University Press, 1993)
Jonathan Harris, Byzantium and the Crusades (Hambledon and London, 2003)
John L. La Monte, trans., The Wars of Frederick II Against the Ibelins in Syria and Cyprus by Philip de Novare (Columbia University Press, 1936)
Lynn Jones and Henry Maguire, "A Description of the Jousts of Manuel I Komnenos" in Byzantine and Modern Greek Studies 26 (2002), pg. 104-148
The Old French Chronicle of Morea: An Account of Frankish Greece after the Fourth Crusade, trans. Anne Van Arsdall and Helen Moody (Routledge, 2018)