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This is ridiculously complicated.
If you're asking if the First Crusade was caused by violence inspired by the Qur'an in a manner similar to modern jihadist violence, the answer is a categorical, "No."
If you're asking if Muslims committed acts of violence that brought the crusades into being, the answer is kind of.
The Muslims in question were the Seljuq Turks, a group from the Asian Steppes who had conquered a good portion of Persia and assimilated Persian culture to a high degree, including converting to Islam. As a steppe people and as Persians, all historical precedent drove them into conflict with the Romans - or Byzantines as we know them. The wars between Byzantium and the Seljuqs were not religious conflicts, but rather purely about up-and-comers establishing themselves in the region. However, the emperor Alexios, having recently lost quite a bit of territory, had the brilliant idea to ask the Christians of Western Europe to help out a fellow Christian. The wars had certainly interrupted Christian pilgrimages, and plenty of atrocities had been committed (as they often were at that time), but there wasn't an intentional pogrom against Christians.
Pope Urban II was quite receptive to Alexios' plea, in part because he saw the Turks as a threat to Christianity, in part because he wanted a more unified Christendom, and in part out of a desire to direct a lot of highly-trained, extremely bored warriors away from killing fellow Christians in the spats that tended to erupt in Europe.
So, yeah, the leaders of Christendom had different goals for different reasons, and the Crusaders themselves probably genuinely believed Christians should rule the Holy Land. Of course, they killed more Christians than the Seljuqs ever did, but that's war for you.
There are a lot of reasons the First Crusade happened, modern scholars posit new contributing reasons seemingly every decade or so. One example: the Crusades were a way for families to get rid of heirs. If you are a lord during that time period and you have seven sons, that means your estate will be divided quite thinly after your death. Scholars have used the few surviving records to suggest that most new recruits to the First Crusade were in fact lesser sons of their families.
Which brings me to how they were recruited. The Church used extremely emotional, rhetoric-filled sermons where they'd get young men to promise to participate on the Crusade before they had time to really consider it. One author painted an image of a young man getting inspired by tales of horrible Muslims tying Christian women to stakes by their intestines before burning them (an actual example of a story used during the First Crusade), then waking up in the morning and confronting the reality that he was now obligated to give up his whole life and likely die marching to a far away land.
So, we know that the Church was using Holy Land rhetoric, not defend Constantinople rhetoric. It stands to reason that the defense of Constantinople was a welcomed excuse for the Pope to amass an army. The leader of Constantinople at the time was reportedly resistant when the Crusaders arrived, drove back the Muslims, and asked for troops to contribute to the march on the Holy Land.The march to the Holy Land had little to do with defending Constantinople, to the extent that later Crusaders actually sacked Constantinople.
The Pope himself was also influenced by lesser cardinals. We have written records of cardinals (or bishops) telling the Pope of Christian churches in the Levant being held hostage and burned by Muslims, and the need to rescue Christendom from this threat.
So, in my experience researching the topic, the main reasons cited for the cause of the First Crusade are as follows. Keep in mind that these are in debate due to the lack of unbiased source material on the First Crusade.
A desire for the Catholic Church to get rid of internal strife and unite Europe under a common religious cause, which would make the Pope the de facto most influential power figure in Europe.
Genuine religious desire to take Jerusalem and genuine religious hatred of Muslims.
Internal conflict in the Muslim world which prevented any type of unified Muslim defense.
Extremely strong and graphic rhetoric employed by the Church.
First Crusade, A New History by Thomas Asbridge.
People of the First Crusade by Michael Foss.
Chronicles of the Crusades by Jean de Joinville
The First Crusade and the Idea of Crusading by Jonathan Riley-Smith
hi! fyi, you'll find lots of great info on the causes of the First Crusade (and others) in the FAQ
Most of the posts have been archived now, so if you have follow-up questions, just post them here & mention the relevant user's username to notify them
As others have said, this is a complicated question. To answer it requires a few centuries of context.
In the middle of the seventh century, Muslim armies made their entry into Byzantine territory, seizing from Syria to Egypt and later across North Africa. For minority Christian sects in these territories, this was actually a good thing. Their new Arab lords allowed them a freedom of religion that the Byzantines had not. Christians lived in relative peace with their new Muslim conquerors for centuries in the Near East.
However, in the eleventh century, the Turks enter into the equation. Turkish forces sweep through Armenia, Syria, and Palestine. Even parts of the Byzantine breadbasket in Anatolia were lost after Emperor Romanus IV's spectacular defeat at Manzikert in 1071. Though Muslims, the Turks were not their Arab predecessors. They were shocked at how prevalent and prosperous Christian communities in their conquered territories were, especially in Jerusalem. In the aftermath of their conquest, Turkish forces razed churches, killed clergymen, and attacked Christian pilgrims, at least until they realized the significance of Christianity in Jerusalem. In short, the pilgrims were profitable enough to tolerate. Even so, the Turkish conquests made travelling to the Holy Land a much more dangerous endeavor than it had been under Arab rule.
The Byzantine losses were well-known in the West. The Normans had pushed the Byzantines out of Italy, and there was pressure from Pechenegs to the north. As early as 1073, Pope Gregory VII was drafting plans for a campaign against the Turks. As a result of two centuries of fighting Hungarians, Vikings, and Muslims, Europe was rife with well-armed and well-trained fighting men. Lacking a common enemy and a central power, this led to violence throughout Europe. The Pope saw this as a three-fold opportunity. Sending these bickering warriors to fight a common enemy would free Europe from its multitude of petty infightings, protect Christendom, and most importantly, gain recognition from the Byzantines.
When Alexios took the Byzantine throne in 1081, he inherited this mess of trouble. His forerunners had lost huge chunks of the Empire to the Turkish invaders and left him facing pressure from all sides. Alexios called for aid aid from the Pope in 1095, breaking with tradition. In the eyes of the Byzantines, the West was rightfully Roman land, occupied by misguided Christian barbarians. In requesting their aid, Alexios was acknowledging their claims of ownership of this land, and recognizing the authority of the Pope over many of these countries. Alexios saw this as an opportunity to gather support from fellow Christians against the growing Muslim threat. Gregory saw it as the beginning of a campaign to assert the authority of the Pope over all of Christendom, especially in light of the Schism in 1054.
On November 27, 1095, Pope Urban II preached for the Crusade to the aid the Byzantines at the Council of Clermont. He continued Gregory's work, declaring that Eastern Christians faced persecution under Muslim rule and that Christendom's eastern half, the Byzantine Empire, was in dire need of assistance. But defending the Byzantines would not be enough to raise the army Gregory had envisioned; he needed to sweeten the deal. So Urban changed his focus. Sweeping through Asia Minor, this Christian army would repel the Seljuk invaders from Byzantine lands and then proceed to liberate the Holy Land. The Byzantine call for aid had sparked a Crusade for Jerusalem.
Depending on how far back you want to go, you can answer either way. In reality, there was no one cause of the Crusades, rather a mixture of causes:
Though the excuses of Turkish persecution of Christians had some grounding, they were dramatically exaggerated for the sake of riling up young men to fight.
Sources: Thomas F. Madden. The New Concise History of the Crusades.
Amim Maalouf. The Crusades Through Arab Eyes. Translated by Jon Rothschild.
TL;DR there is some legitimacy to the idea that aggression from the Turks, who were Muslim, contributed to the First Crusade, but in reality, there are political and religious motivations from Europe and Byzantium that were also major factors.