Janus gave January its name because he was the Roman god associated with doors, transitions and beginnings.
The month derives from the Latin name Ianuarius, meaning Janus's month. Janus was commonly depicted with two faces looking in opposite directions, a later symbol of looking back at the past while facing the future. That image makes the deity especially fitting for the turn from one year to the next.
January was not always the first month of the Roman calendar. In the early Roman tradition, March was the starting month, but January and February were later placed before it. January became the first month in the Julian calendar reforms associated with Julius Caesar in the first century BCE, and it retained that position in the Gregorian calendar used today.
Janus is not the same figure as Saturn, whose festival Saturnalia occurred in December. The name of January reflects transition and beginnings, not the Roman god of agriculture.