A polynomial \(D\in {\mathbb {Z}}[x]\) is called Pellian over \({\mathbb {Z}}\) if the polynomial Pell equation \(P^2-DQ^2=1\) has a non-trivial solution in \({\mathbb {Z}}[x]\) . It is an open problem to determine all the polynomials \(D\in {\mathbb {Z}}[x]\) that are Pellian over \({\mathbb {Z}}\) . In the literature, the study of the Pellian polynomials over \({\mathbb {Z}}\) has been mostly restricted to monic polynomials as there are many key difficulties involved in the non-monic case. In this article, we overcome those difficulties and characterise all the quadratic polynomials in \({\mathbb {Z}}[x]\) that are Pellian over \({\mathbb {Z}}\) by providing a necessary and sufficient condition for Pellianity over \({\mathbb {Z}}\) . This seems to be the first instance of a comprehensive study on the Pellianity over \({\mathbb {Z}}\) of non-monic quadratic polynomials in \({\mathbb {Z}}[x]\) . A key difficulty is to find solutions in \({\mathbb {Z}}[x]\) , when the necessary condition is satisfied. Our proof exhibits a constructive method to do so.
