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In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was used by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931)

The natural numbers, N, form a commutative monoid under addition (identity element zero), or multiplication (identity element one). A submonoid of N under addition is called a numerical monoid.
The positive integers, N ∖ {0}, form a commutative monoid under multiplication (identity element one).

The set of integers Z forms a commutative ring under addition and multiplication

What happens if I replace a unique natural number that form a commutative Monoid with the set of integers Z that form a commutative Ring ?

The more specific question should be: What happens if I use (not if I replace) a unique natural number that form a commutative Monoid as (not with) the set of integers Z that form a commutative Ring ?

Monoid $Rightarrow$ Ring

P.S: I don't want replace formally but I do not want to really change, but to make it believe that it is as if I had replaced.

The purpose of Gödel coding is the process of arithmetization, by which syntactic or other combinatorial ideas that are not directly expressible in arithmetic can be simulated by a process of faithful representation. The point is that in our arithmetic theories, we are entitled to refer to concepts such as formulas and proofs and so on, by means of the Gödel codes.

Meanwhile, the structure of the natural numbers $langlemathbb{N},+,cdot,0,1rangle$ is bi-interpretable with the structure of the integers $langlemathbb{Z},+,cdot,0,1rangle$, in the sense that each of these structures can define a copy of the other, in such a way that it can also definably see how it is translated into the others. Namely, the natural numbers $mathbb{N}$ are the sums-of-four-squares in $mathbb{Z}$, and there are diverse ways to define a copy of $mathbb{Z}$ inside $mathbb{N}$.

So although it is traditional to take the Peano axioms PA as an axiomatization of the natural numbers $mathbb{N}$, one can translate the axioms to make a equivalent corresponding axiomatization of $mathbb{Z}$.

In this sense, it doesn't matter whether you use natural numbers or integers in the Gödel coding; doing so will achiever the same ends in the end. There is essentially no fundamental difference in the ideas and methods.

Suppose you were trying a similar attempt of arithmetization, but you decided to do so for Presburger arithmetic. Well, there would be a couple of problems. The first is that you have only addition, so that your possibilities for encoding are limited, and it would be challenging to find a unique code for every statement, AND be able to refer to this code inside the theory. The second is that Presburger arithmetic is decidable, so you cannot get an incompleteness result like that of Goedel: you can tell which statements are true of the (additive) monoid of the natural numbers.

The reason Goedel chose (the semiring of) natural numbers is because the theory he was working with had as a model the natural numbers with multiplication and addition. If you chose to axiomatize properties of another structure such as a commutative ring, you could try to use the language to code statements by members of the structure, but you could not necessarily get the results Goedel did; it depends on whether the theory of the structure is (inherently? I forget the technical adjective) undecidable.

Gerhard "The Other Hand Has Fingers" Paseman, 2019.01.02.