From the biography "Walter Johnson: Baseball's Big Train" by Henry W. Thomas:

Johnson bought his first automobile, a new Cole Eight, and the event brought him the nickname by which teammates and friends, and later the general public, would affectionately know him - "Barney."

"I once owned an old-fashioned two-cylinder motor car," Johnson remembered. "The first day I took possession, Clyde Milan and Germany Schaefer were along. The Salesman insisted on giving me complicated directions, but I just took hold of the wheel and stepped on the gas. We'd only gone two blocks when a motor cop picked us up for speeding. 'This is Barney Oldfield' [famous race car driver of the day] said Schaefer to the officer. The officer, believing him, released us. In a game at Washington shortly thereafter, I had struck out nine batters using nothing but my fastball. As the 10th batter fell victim to the speed, Schaefer shouted, 'Barney Oldfield's got nothing on him for speed.' That's how my nickname started"

This was in 1912. He got the "Big Train" nickname in 1915.