For example: "It takes four quarters to complete a football game." Is that mean" "It takes one hour to complete a football game."
In general, a quarter means 1/4.
Sports: In sports, "quarter" means 1/4 of the game, and it does not necessarily mean 15 minutes. Four quarters means the game is divided into four equal "parts" with some type rules between them - such as providing the players a short break. The duration of each quarter depends on the official rules for the game. For example from the Wikipedia article about basketball, a quarter could be 10 minutes (FIBA), 12 minutes (NBA), or 8 minutes (United States high school varsity games).
Quarters in sports are typically broken up into four parts by time. But it's also possible for a game to be broken up into quarters based on some aspect other than time. (Though I don't have any example of that.)
Time: "Quarter" is often used in describing 1/4 of an hour (15 minutes), 1/4 of a year (every 3 months, such as "Quarterly Reports" issued by American corporations), and 1/4 of a century (25 years).
The title of your question is generically asking about "four quarters". There are many other meanings of "quarters" as well. In America, notably, "four quarters" is one dollar.
US football is played in quarters, each of which has fifteen minutes of actual play, with a short break after the first and third quarters, and a longer break ('halftime') after the second.
However, unlike what the rest of the world calls football (and Americans call soccer) there are frequent pauses in play which are not counted 'on the clock', so the entire time required by a full game is around two and a half hours—even longer when the game is televised and more breaks are taken so the broadcaster can play more commercials.