Is it possible for a number to be a rational number that is not an integer but is a whole number?

No it is not possible. If a number is a whole number, then it is automatically an integer. For example, the number 5 is a whole number. It is also an integer

Set of whole numbers = {0, 1, 2, 3, 4, 5, 6, 7, 8, ...}
the three dots indicate it continues on forever in that pattern

Set of integers = {..., -3, -2, -1, 0, 1, 2, 3, ... }
Basically the set of integers is the set of whole numbers with negative whole numbers thrown in as well