Relations between Middle Eastern nations and the Kingdoms of Christendom was at an all time low at this period, and Byzantium in particular was particularly hostile towards Western Crusaders following the debacle in the Fourth Crusade https://en.wikipedia.org/wiki/Fourth_Crusade, which despite the Pope's explicit instruction not to aggregate Alexis IV saw Constantinople sacked, the emperor killed and a pretender put on the throne. In return, crusaders took huge amounts of loot, and Venetian traders received huge amounts of trading power. Byzantium slowly fell into decline after this period.

Muslim relations were little better, but rarely showed themselves to be openly hostile in terms of violence in the interlude between Crusades, with the exception of the Reconquista in Spain, which can technically be classified as a crusade due to it's endorsement by the Papacy in the form of mandates and legates. In the Levant, on the ground however, trade and relation flourished between the common people and merchants of all races and religions; on a religious and political level things were more hostile. With the exceptions of minor fighting between holy orders and local radicals, the Kingdom of Jerusalem did ally itself with Muslim leaders throughout it's history, but this was more on a local level. The Mamelukes and the various Sultanates were hostile, not always to the point of war, but were hostile nontheless

I'm taking this from the excellent book on the period "God's War: A new History of the Crusades" by Christopher Tyerman which I can't recommend enough