There are 4 quarts in a gallon. That holds for both the US gallon and the imperial (UK) gallon, since each is divided into four quarts, eight pints, or 32 gills.
The gallons themselves differ in size: a US liquid gallon is 128 US fluid ounces (about 3.785 liters), while an imperial gallon is about 4.546 liters, so a US quart is smaller than an imperial quart.
Source: Wikipedia, Gallon