Not every counting number is an integer true or false?

False. All counting numbers {1, 2, 3, 4, 5, ...} are integers. The set of integers is {..., -3, -2, -1, 0, 1, 2, 3, } which is basically the set of positive whole numbers, negative whole numbers and 0. 

Note: the set of counting numbers is a subset of the set of integers.