You just say G(x) = f(x+1) - f(x)

G(x) is a polynomial, not a set. And for specific values of x, G(x) is a number.

So G(x) = f(x+1)-f(x) represents both the statement that the polynomial G(x) is the same as the polynomial f(x+1)-f(x) and the statement that the number G(x) is the same as the number f(x+1)-f(x) for every number x.

You can also use the delta operator ∆ to denote the same thing. For any function f(x), not just polynomial, ∆f(x) means f(x+1)-f(x). And then you can use that to denote the finite difference of a finite difference. The thing you wrote above as G'(x) could be written as ∆^(2)f(x)