In the 1st Century BC, the Romans conquered or absorbed the remnants of the Greek kingdoms in the Middle East, including Israel. They inherited the Greek established governing institutions. Additionally, Greek culture had a tremendous influence on the Romans, especially the elite. Not unlike the old Eastern WASP Establishment that mimicked the private boarding schools, the architecture, the clothing styles, and even the pronunciation of the British upper classes, the Roman upper classes were enthralled with all things Greek. To a Roman aristocrat, a post in government service in the Greek-influenced Middle East was far more prestigious than a similar one in then-barbarous western Europe.
In addition, there might have been some trickle down effect on upwardly mobile people, such as the centurion. Furthermore, it is possible his unit was drawn from an area like Asia Minor, where Greek was the lingua franca.
However, neither Scripture nor other sources provide a clear answer as to whether Jesus Christ spoke koine Greek.