I don't know of any quantitative studies into this question, which makes it difficult to answer, but I would be inclined to argue that the answer is "no, the Middle East is not more conflict prone than other regions." There are a few things that might give it that impression. For one thing the Middle East is the "cradle of civilization" so if you go back thousands of years the Middle East is pretty much the only region of the world that has a history of conflict to study while the rest of the world remained pre-historic.

Other conflicts can give the impression of being old but are newer than you might think. While the Shia-Sunni conflict in the 21st century can, in some senses, be traced back to Muhammad's succession in the 7th century, the fact is that the Shia-Sunni conflict, historically, hasn't resulted in all that much bloodshed. Likewise, while we can look at and discuss some of the centuries old examples of Muslim/Jewish violence, but it's not really until the first-Zionist migrations in the late 19th century, and especially the post-Balfour declaration migrations, that that conflict starts to resemble the current Arab/Israeli issues we see today.

Likewise I could point to counter examples. Pax Ottomana kept most of the Middle East relatively peaceful at a time when Europe was continuously slaughtering itself.

I don't think it would require me to go very far out on a limb to suggest that it's far more likely to be Europe, and not the Middle East, that at least prior to 1945 seems to have been the world-wide leader in creating mass violence.

As for why the late 20th century and contemporary Middle East seems to be a particularly violence prone region? I would point to the creation of nation-states, which everywhere since the rise of European nationalism has been a profoundly violent process, as being something that didn't really happen organically in the Middle East. Nationalism itself became a powerful ideology only in the 20th century, which also saw the rise of various and competing nationalisms (anti-colonialism, pan-Arab nationalism, pan-Islamic nationalism, "local" nationalisms, socialism, Zionism, anti-Zionism, tribalism, and so-on) that today seem to be sorting themselves out as such disputes have historically sorted themselves out in other places like Europe, with ethnic cleansing, war and mass-violence.