I don't know how to define in the logic of the first order the following statement: "The set of natural numbers $N$ is closed with respect to the sum operation between them".
For this purpose it’s possible to use the operator + in the formulas, in addition to existential and universal quantifiers and logical operators.
Are you thinking of a first-order theory of natural numbers such as PA, or of sets such as ZF? For a first-order theory of natural numbers you could write $\forall a\forall b\exists c(c=a+b)$, whereas for a first-order theory of sets you could write $\forall a\in\Bbb N\forall b\in\Bbb N(a+b\in\Bbb N)$ (or even $\forall a\in\Bbb N\forall b\in\Bbb N\exists c\in\Bbb N(c=a+b)$).