No, it's not true. One can say that the Middle East has witnessed conflict for millennia, but that is true of every part of the world. Few of the current states existed even one hundred years ago. Syria, Iraq, the Hejaz and Turkey were part of the Ottiman Empire in 1916 (and each would become a seperate territory at the end of the First World War). Iran was ruled by the Qajars and had totally different priorities then, and anyway was convulsing in civil conflicts which would eventually see the Pahlavis come to power. There was a local power in the central desert, the Sultanate of Nejd, led by a man called Ibn Saud, who would eventually conquer the Hejaz (where Mecca is) and form Saudi Arabia in the 1930s. Regional politics post-WWI (and conflict) was defined largely by factors irrelevant to the present except as a legacy. They include colonialism: civil conflicts in Syria and Iraq were between local people against the French/British (it gets more complicated than that, but I'm trying to paint broad strokes).

There isn't very much of today's issues which stretches back further than about 40-50 years (that even includes the rise of Saddam Hussein, of Hafez al Assad, the rise of Saudi Arabia as a major regional player, the Iranian revolution, the independence of the Gulf Emirates from the British Empire, Egypt's peace with Israel and other factors too). Of course previous legacies are important, but they do not factor politically in any meaningful way.

In no way does the conflict between Saudi Arabia and Iran have anything to do with the Ottomans and their conflict with Iran. It certainly has nothing to do with the wars of the Sassanians and Rome. Iran's Shia faith is not in a zero sum war with the Sunni Islam - Iran wasn't even Shia until 600 years ago, and religioun has more often been used to justify conflicts than to provoke them. It certainly has nothing to do with the Arab conquest of Persia 1400 years ago, although propaganda on both sides of the Iraq-Iran war may have claimed so.

It is a completely illogical argument not rooted in reality, and there isn't any more nuanced answer than to say no, read the history and use your critical thinking, and you'll see you can't pin today's conflicts on pre-20th century (let alone ancient) legacies and history.