First, you're looking for beasts of the form
Second, a polynomial of degree 3 or less over any field is irreducible iff it has no roots in the field, so you want that the above quadratic's discriminant has no solution in our field, i.e. : you want that is NOT a quadratic residue .
Well now, just check what elements in are not quadratic residues and count up all the possibilites for ...
For example, as , we get that 5 is not a quadratic residue mod 13 ==> every pair of elements in this field will yield an irreducible quadratic, for example:
** is one irreducible quadratic;
** the quadratic is irreducible...etc.
Of course, there is a formula, but I really don't remember it, though you can look for it in the books...