Is this correct: "一个游戏有四个季度"
No. 

A quarter in a sports game is usually translated as 节. For example, fourth quarter is 第四节. The first and second quarters are collectively referred to as 上半场 and similarly, the third and fourth quarters are referred to as 下半场. A sports season is usually translated as 赛季．

季度, when tranlated into English, only refers to a quarter of a year, as you have noted.