It is an open problem. Maximal results about bijections from
$\mathbb N\times \mathbb N$, $\mathbb Z\times \mathbb N$, $\mathbb Z\times \mathbb Z$ to $\mathbb N$ are contained in

John S. Lew, Arnold L. Rosenberg,
*Polynomial indexing of integer lattice-points I. General concepts and quadratic polynomials*, J. Number Theory **10** (1978) pp 192-214, doi:10.1016/0022-314X(78)90035-5.

*Polynomial indexing of integer lattice-points II. Nonexistence results for higher-degree polynomials*, J. Number Theory **10** (1978) pp 215-243,
doi:10.1016/0022-314X(78)90036-7