Determining the polynomials $D \in {\mathbb Z}[x]$ such that the polynomial Pell equation ${P^2-DQ^2=1}$ has nontrivial solutions $P,Q$ in ${\mathbb Q}[x]$ (and in ${\mathbb Z}[x]$) is an open question. In this article, we consider the generalized polynomial Pell equation $P^2-DQ^2=n$, where $D \in {\mathbb Z}[x]$ is a monic quadratic polynomial and n is a nonzero integer. For $n=1$, such an equation always has nontrivial solutions in ${\mathbb Q}[x]$, but for a non-square integer n, the generalized polynomial Pell equation $P^2-DQ^2=n$ may not always have a solution in ${\mathbb Q}[x]$. Depending on n, we determine the polynomials $D=x^2+cx+d$, for which the equation $P^2-DQ^2=n$ has nontrivial solutions in ${\mathbb Q}[x]$ and in ${\mathbb Z}[x]$. Taking $n=-1$, this allows us to solve the negative polynomial Pell equation completely for any such D. An interesting feature is that there are certain polynomials D for which the generalized polynomial Pell equation has nontrivial solutions in ${\mathbb Z}[x]$, but only finitely many, whereas the solutions in ${\mathbb Q}[x]$ are infinitely many. Finally, we determine the monic quadratic polynomials D for which the solutions of $P^2-DQ^2=n$ in ${\mathbb Z}[x]$ exhibit this finiteness phenomenon.