I don't believe there has ever been an all out war fought purely because of a theological dispute about the trinity. However, trinitarian doctrine was one of the more contested parts of early Christian theology and still is a major point of contention between the Eastern and Western Church.

In the Late Antique and medieval periods, disputes about the trinity were some of the biggest issues Christians argued about. To be orthodox as opposed to heterodox, as defined by the doctrines developed in the first three to five hundred years of Christian thought, one of the most important beliefs was the correct understanding of the triune nature of God. Orthodoxy meant believing in a single godhead made of three equal parts, the Father, the Son, and the Holy Spirit. A variety of heresies developed which asserted different versions of this. These include unitarianism, which rejects the trinity, Arianism, which posits that the Son is subordinate to the father, adoptionism, literally that the Father adopted Jesus, modalism, monarchism, etc. Arianism which was originally formulated by a theologian called Arius, was probably the most prominent Trinitarian heresy.

In fact, many "barbarian" groups were converted, but became Arians, which required the orthodox church to take a fair amount of action to protect trinitarian doctrine. This may be one source of confusion for you.

Another conflict involving Trinitarianism arose between the Eastern and Western churches during the Middle Ages. It's a bit complicated, but essentially Eastern theologians rejected the insertion of Filioque (and the Son) into the creed. The actual theological implications are obscure, and I can go into them more if you'd like. But, ultimately this, coupled with a few other theological and ecclesiological issues, resulted in the Schism, that is the splitting of the Eastern and Western churches. This was non-violent, though.