By "Church HQ," I assume you mean Rome.

Before the advent of Islam and the Muslim conquests of the seventh century, when the entire Mediterranean basin was Christian, the bishops of five cities were recognized as patriarchs - men whose authority extended over entire regions. The most important of these bishops presided at Rome, and was already beginning to be called "pappa" (pope). The other four were based at Constantinople, Antioch, Jerusalem, and Alexandria. This system, which had evolved over several centuries, reflected the fact that early Christianity's center of gravity was in the eastern Mediterranean. It was only the subjection of this area to Muslim rule, and the isolation and mass conversion to Islam that followed, that made Rome so much more visible than the other four ancient patriarchates.

The members of the Jesus movement, famously, were first called Christians in Antioch, and for the first three centuries of its existence, the great majority of Christians lived in the Greek-speaking eastern provinces. Thanks to its sheer size and prominence, however, the city of Rome became a prominent Christian center very early. There was already a substantial community by the reign of Nero, who is said to have accused Christians of starting the Great Fire of Rome in 64 CE (Tacitus, Annals 15.44). And by the mid-third century, the authority of the bishop of Rome was starting to be recognized well outside the city itself.

The evolution of the church hierarchy was gradual, and only given definitive form at the Council of Nicaea in 325. But usually, the most important bishops were those who presided over the late Roman Empire's most important cities. Over the fourth century, in fact, the ecclesiastical hierarchy came to mirror the administrative hierarchy quite neatly. Politics, however, forced some adjustments. The rapid rise and political importance of Constantinople had to be acknowledged, and an ambitious bishop of Jerusalem managed to ensure that his city was also recognized as the seat of the patriarchate. The Council of Chalcedon (451 CE) established the definitive arrangement of five patriarchs. In this system, Rome was supreme over the churches of Italy, and (since no other city in western Europe was nearly as prominent) had implicit moral authority over a huge part of the Empire.

Despite the fact that many of the Germanic tribesmen who assumed political control of western Europe in the fifth century were Arian heretics, the system of five patriarchs remained more or less stable until the Muslim conquests placed Antioch, Jerusalem, and Alexandria under non-Christian political authorities. Although patriarchs continued to be chosen, the fact that they were isolated from the rest of Christendom (and, eventually, the conversion of much of their flock to Islam) considerably lessened their real authority. The patriarch of Constantinople, of course, remained very influential (if often subject to imperial interference) for centuries longer. The decline and fall of the Byzantine Empire, however, eventually made this patriarch and his flock subjects of the Muslim Ottoman Empire. And so the pope of Rome -already the most important Christian bishop - was left as the last patriarch in a Christian-ruled country.

Edit: When I answered this question, I assumed that the OP meant "why is the headquarters of the largest Christian denomination not in the Middle East (i.e., at Rome)?" The "HQs" of several very ancient Christian churches are of course in the Middle East.

In response, finally, to the excellent comments of /u/ThreeHornedSnail, I should clarify that my focus on the formal church hierarchy in this answer is only a means of tracing how Christendom's political/symbolic center of gravity (at least in western European eyes) shifted westward in late antiquity and the middle ages.

Well, as a Catholic I am of course honoured if you refer to Rome as "Church HQ" (or at least I assume so), but of course there is a vast number of Christian churches and communities with their respective headquarters all over the world (some of them even in or near Jerusalem) and many others even not having "headquarters" at all.

So, why Rome and not Jerusalem? The exact reasons behind this are complicated, sometimes not all that clear after nearly two millennia and also still a point of contention between various Christian churches. The short version which also happens to follow Catholic doctrine goes as thus: Christ said to Peter: "And I tell you that you are Peter, and on this rock I will build my church, and the gates of Hades will not overcome it." So Peter is the focal point of the emergent Christian church, and when he travelled to Rome, became bishop there and eventually got martyred, his office was inherited by his successors in Rome. So far for the official Catholic version.

The historical truth is hard to pin down here, it is indeed not even proven (and probably not provable at all) whether Peter ever was in Rome at all. This is also not my area of expertise at all, so I hope someone else more knowledgeable will be able to follow up on my answer or correct me if necessary. The longer version of "why Rome?" goes thus: When the early Church started to develop an internal structure and hierarchy, eventually five episcopal sees emerged who were considered the most important. Those were the sees of Jerusalem, Antioch, Alexandria, Constantinople and Rome. As you can see, all of them were located within the borders of the Roman Empire at the time, and with the exception of Rome they were all part of the Eastern Empire to boot. Jerusalem was of course the place were much of Christ's ministry took place, Antioch and Alexandria had flourishing Jewish and Christian communities and belonged to the largest and most important cities in the Empire overall, while Constantinople and Rome where the seats of imperial and political power. Due to this and because of the St Peter's legacy, Rome was considered to be the "first among equals" in a ceremonial order of precedence ranking those five sees.

This arrangement (commonly known as "pentarchy") was codified relatively late and is closely tied to the rise of Christianity within Roman society, going from an obscure branch of a small minority religion to the official religion of the Empire. The order of precedence I mentioned above was, as I said, purely ceremonial; in theory Rome had no special power over the other sees. Yet the close proximity of the bishops of Rome and Constantinople to the imperial residence(s) naturally led to them having a better connection to the Emperor, which translated to real political power after Christianity became the Roman state religion. There was no separation of church and state there; instead, the various Emperors actively took part in church politics and would often intervene when theological conflicts arose, further strengthening the de facto power and preeminence of those two sees, who had a much easier time lobbying the imperial court and following imperial politics.

There were other factors too, of course. Two very important ones are the slowly but steadily growing differences (culturally, theologically, politically) between the Eastern and the Western half, or in case of the pentarchy: between Rome as the sole western patriarchate and the other four eastern sees. This was even a problem of language, since the eastern half spoke Greek (or other languages like Egyptian or Aramaic), while the Roman church was speaking Latin. The other problem was that first the military expansion of the Sassanide empire and then the rise of Islam and its quick spread meant that many of those Christian communities especially in northern Africa and the Middle East were lost to the Roman Empire, and were ruled by non-Christian monarchs now. This in turn meant that those communities who now were diaspora churches were forced to align themselves even more closely with Constantinople, whose patriarch now sat solely at the seat of power and was in the eyes of the Emperor the main representative of the Christian faith, as the city of Rome had lost much of its economical and political power over the years and now also found itself not in the centre but the periphery of the Empire, with the imperial grasp and presence in Italy becoming more and more tenuous as the centuries went on. Despite that, the Bishops of Rome hadn't forgotten the legacy of St Peter and continued claiming the top spot withing the Church for themselves, slowly but surely upgrading from "first among equals" to "first, full stop". Political quarrels between the Emperor in Constantinople and the emerging landscape of Germanic kingdoms in Western and Central Europe (eventually culminating in Charlemagne styling himself Emperor) further contributed to the growing gap of communication and understanding between East and West.

So there you have it: Rome being the centre of the largest Christian Church is mostly the result of political developments millennia ago, and sometimes also sheer luck. Rome was the sole part of the pentarchy in the west, m and while first the cities Jerusalem, Antioch and Alexandria and ultimately also Constantinople were conquered by Muslim forces with their respective Christian communities losing their political edge and ultimately many of their members too, Rome was able to assert itself as the sole successor to Peter as the "Lord of the Apostles" in a western part of Europe that never became (fully, at least) part of a non-Christian empire.

I said it before: This is a vast and very complicated topic, and I am well out of my expertise here, so I really hope that others will step up and expand on what I wrote here! :-)