When did the Church in the West begin communicating in Latin, instead of Greek? Since the Gospels, Paul's epistles, and the other books of the New Testament were written mostly in Greek, as well as much writing of early Church fathers, it stands to reason that much – if not most – of the early Christian scholarly writings were in the Greek language. In addition, the languages that were spoken by those Christians are Syriac and Greek. 

By the early middle ages, however, it is clear that nearly all scholarly writing in the West was done in Latin. 

When did Christian scholars in the West begin writing and publishing works primarily in the Latin language, instead of the Greek language? Why?
The first major Christian theologian to write primarily in Latin was Tertullian (155–240): his association with a heretical movement is all that prevents him from being universally considered the first of the "Latin Fathers." Jerome provides a brief summary of his early life:


  Tertullian the presbyter, now regarded as chief of the Latin writers after Victor and Apollonius, was from the city of Carthage in the province of Africa, and was the son of a proconsul or Centurion. (On Illustrious Men, chapter 53)


Carthage was a Roman city at the time, so Latin would have been Tertullian's native language.  As a result, it's not surprising that he would write in that language, not just for his own comfort but also to reach an audience that did not speak Greek.

Perhaps the most important cause of Latin's dominance in the Western church is Constantine's conversion in the 4th century and the subsequent official status that Christianity enjoyed in the Roman Empire.  Latin served as the Western Empire's lingua franca, and the dominance of the language continued in Europe even after the Western Empire declined and finally fell in the 5th century.  

In the Greek-speaking Eastern Empire, Greek fathers like Maximus the Confessor and John of Damascus wrote into the early medieval period.  Their influence on the Western church was perhaps not as great as that of Latin fathers like Jerome, Augustine and Gregory the Great, who were more associated with, and geographically close to, Rome.  But it was Great Schism, beginning in the 11th century, that ultimately resulted in Greek-speaking populations disassociating themselves from the Catholic Church, leaving Latin without a significant competitor.