Remember, that's properly "The Christ". There were sects in Israel/Palestine/WhateverHaveYou well before the (totally unproven) birth of "Jesus The Christ" who described themselves as "Christians".
This was a time of tremendous social and religious termoil and the religion that became Christianity was a past-up of various ideas that were going around at the tiem.
There is no actual evidence of a historical Jesus in any way that hasn't been shown to be a forgery. Roman historians who railed against the Christians do not mention this Jesus person at all. The Jesus story seems not to appear until around 100 years after the "the birth of Christ".
For certain the dates don't make sense because Roman tax time was logically in the spring, as it still is today, and there is no record of any need to travel to ones birthplace "to be counted", or, in fact, of any count at all (and examination of the court records of Pontius Pilate showed no evidence of the crusifixion).
This does not in any way invalidate the religion or its teachings. In the custom of the time the actual existance of Jesus as a living person would have been completely irrelevant. Remember, these were Pagan times, and gods and godesses were not actual (and not worshiped), they were personifications of natural and social forces, personifications that allowed relationships between forces to be stated in human terms understandable to people without formal education.
Therefore the actual historical existance of Jesus "The Christ" is entirely irrelevant. Worrying about this is an artifact of our current literally minded era. It simply doesn't matter.
So yes, there were Christians well before the "birth of 'The Christ'".