No. One, a counting number, doesn't belong to either of those sets.
The set of counting numbers is a proper subset of the whole number. The latter includes negative counting numbers. Also, there is no consensus as to whether 0 belongs to counting numbers or whole numbers.
The set of counting numbers is denoted by N.
The immediate [next] superset is, trivially, the set of natural numbers which consists of the counting numbers and zero. The next significant superset is the set of integers: the counting numbers, their additive inverses (or negatives) and zero.
This set of numbers is called "Whole Numbers".
The set of Whole numbers includes the set of counting numbers and Zero.
By definition, the set of counting numbers starts at one and proceeds in ascending order. The next number is 2. If two were not the next number in the set, it would not be the set of counting numbers.
The set of Natural numbers.
The set of counting (natural) numbers is the set of all positive integers, while the set of whole numbers is the set of all positive integers included zero.