Did Christianity originate in Ethiopia?

No. Christianity originated in Jerusalem, probably in either
AD 29 or AD 33 (the two years which have the Passover fall on the right
day to fit the description in the gospel accounts). To be specific,
Christianity as a publicly proclaimed religion, began on the Day of
Pentecost, forty-nine days after Jesus was resurrected from the dead.
This is recorded in Acts chapter two.

To be honest, I am not sure where you might have gotten the
idea that Christianity originated in Ethiopia. That is a new one for me.
It is true that Christianity spread to Ethiopia at a very early date.
Acts 8:25-39 records the conversion of a high Ethiopian official who then,
presumably, went back to Ethiopia where he shared about Jesus the Christ.
This happened some time around AD 40; probably ten years or less after the
church had begun in Jerusalem. Some believe the conversion of the
Ethiopian official was the historical beginnings of the church there,
although it cannot be proven. What we do know from historical records is
that Christianity had been established in Ethiopia by about AD 300. The
first Ethiopian “bishop,” Frumentius, was consecrated some time around AD
325. Even today, the national church in Ethiopia is the Coptic church,
which can trace its roots all the way back to the earliest history of
Christianity. The first country to become officially “Christian” was
Armenia, under Tiridates III (A.D. 238-314), who was the first ruler to
Christianize his people.

John Oakes, PhD