Was the first major league game in 1905?

No. The origins of both the American League and National League are from years before 1905... The National League of baseball freanchises (and MLB in general) originated in the late 1800's while the American League of Professional Baseball Franchises was founded in 1901. The first major league game was played May 4, 1871 between the Cleveland Forest Cities and the Fort Wayne Kekiongas. Click on the 'FIrst MLB Game' link on this page to read about it.