A quarter is another way of saying one fourth, in this setting, one fourth of a year = three months. Quarterly means something that happens every 3 months. Edit
Outfielder Ty Cobb, whose career ended in 1928, has the highest batting average in Major League Baseball (MLB) history. He batted .366 over 24 seasons, mostly with the Detroit Tigers. In addition, he won a record 11 batting titles for leading the American League in BA over the course of an entire season.
The first batting average champion in the NL was Ross Barnes; in the league's inaugural 1876 season, Barnes batted .429 for the Chicago White Stockings. The American League (AL) was established in 1901, and Hall of Fame second baseman Nap Lajoie led that league with a .426 average for the Philadelphia Athletics.