I think it's not true in a very clear sense: you can construct semantic models of the axioms of arithmetic in which it is false. In other words, you can construct semantic models of the axioms of arithmetic (more precisely, of the axioms of arithmetic using first-order logic, which is what the article is about) in which there is a "number" that is the Godel number of a proof of the Godel sentence G! Godel's Theorem just ensures that in any such model, the "number" that is the Godel number of such a proof will not be a "standard" natural number, i.e., one that you can obtain from zero via the application of the successor operation.
So another way of viewing this whole issue of "truth" is that it is always relative to some semantic model. If your chosen semantic model of the axioms of arithmetic is the "standard" natural numbers, and only those numbers, then the Godel sentence G for that system will be true--there will indeed be no number in your model that is the Godel number of a proof of G. But if you pick instead a non-standard model that includes "numbers" other than the standard natural numbers, but still satisfies the axioms, then the Godel sentence G can be false in that model, since there can exist a "number" (just not a standard one) in that model that is the Godel number of a proof of G.
What you are saying is true (as you allude to with first-order logic) in any context in which Godel's completeness (not incompleteness) theorem holds. Not all logics have versions of Godel's completeness theorem (e.g. second order logic with full semantics). You can argue that philosophically in systems where Godel's completeness theorem fails that the article's statement is valid. But yes that's why it's only true in "certain senses."
More generally the philosophical question of truth revolves around whether there is a single, true set of standard natural numbers that corresponds to reality, and therefore all nonstandard natural numbers are "artificial" in some sense, or whether even standard natural numbers exist only in a relative sense.
I think it's not true in a very clear sense: you can construct semantic models of the axioms of arithmetic in which it is false. In other words, you can construct semantic models of the axioms of arithmetic (more precisely, of the axioms of arithmetic using first-order logic, which is what the article is about) in which there is a "number" that is the Godel number of a proof of the Godel sentence G! Godel's Theorem just ensures that in any such model, the "number" that is the Godel number of such a proof will not be a "standard" natural number, i.e., one that you can obtain from zero via the application of the successor operation.
So another way of viewing this whole issue of "truth" is that it is always relative to some semantic model. If your chosen semantic model of the axioms of arithmetic is the "standard" natural numbers, and only those numbers, then the Godel sentence G for that system will be true--there will indeed be no number in your model that is the Godel number of a proof of G. But if you pick instead a non-standard model that includes "numbers" other than the standard natural numbers, but still satisfies the axioms, then the Godel sentence G can be false in that model, since there can exist a "number" (just not a standard one) in that model that is the Godel number of a proof of G.