  For a finitely generated group $G$, the \emph{Diophantine problem} over $G$
is the algorithmic problem of deciding whether a given equation
$W(z_1,z_2,\ldots,z_k) = 1$ (perhaps restricted to a fixed subclass of
equations) has a solution in $G$. In this paper, we investigate the algorithmic
complexity of the Diophantine problem for the class $\mathcal{C}$ of quadratic
equations over the metabelian Baumslag-Solitar groups $\mathbf{BS}(1,n)$. We
prove that this problem is $\mathbf{NP}$-complete whenever $n\neq \pm 1$, and
determine the algorithmic complexity for various subclasses (orientable,
nonorientable etc.) of $\mathcal{C}$.
