Let f(x,y) be a polynomial of total degree 2.
Let f_2 be its homogeneous part of degree 2.
It is easy to show that if f_2 is not positive definite,
then f(Z x Z) cannot equal N.
So suppose that f_2 is positive definite.
For a set of primes p of density 1/2, f_2 is irreducible mod p
(this follows from quadratic reciprocity, for instance).
For all but finitely many of these p, the values of f will omit
p-1 residue classes mod p^2 (to see this, observe that over Z/p^2 Z one
can change variables to diagonalize the quadratic form f_2
and complete the square to put f mod p in the form ax^2+by^2+c,
and the irreducibility assumption implies that whenever ax^2+by^2+c
equals c mod p, it also equals c mod p^2).
For such a p, the values of f do not cover all the residue classes
mod p^2, so they certainly don't cover all of N.