The pairing function f(a,b) = (a + b)(a + b + 1)/2 + b is the one that arises by drawing diagonals on the natural number lattice, and marching up them from lower right to upper left. See the picture here.
In that book, it is noted that Polya has proved that any surjective polynomial pairing function is equal to this function or to its dual form f(b,a). (And someone gave a talk here at CUNY a few weeks ago on precisely this fact.) So if this function is not acceptable to you, then you will find no polynomial surjective function.