  We introduce a new family of superalgebras, the quantum walled
Brauer-Clifford superalgebras ${\mathsf {BC}}_{r,s}(q)$. The superalgebra
${\mathsf {BC}}_{r,s}(q)$ is a quantum deformation of the walled
Brauer-Clifford superalgebra ${\mathsf {BC}}_{r,s}$ and a super version of the
quantum walled Brauer algebra. We prove that ${\mathsf {BC}}_{r,s}(q)$ is the
centralizer superalgebra of the action of ${\mathfrak U}_{q}({\mathfrak q}(n))$
on the mixed tensor space $\mathbf{V}_{q}^{r,s}=\mathbf{V}_{q}^{\otimes r}
\otimes (\mathbf{V}_q^*)^{\otimes s}$ when $n \ge r+s$, where ${\mathbf
V}_{q}=\mathbb{C}(q)^{(n|n)}$ is the natural representation of the quantum
enveloping superalgebra ${\mathfrak U}_{q}({\mathfrak q}(n))$ and
$\mathbf{V}_q^*$ is its dual space. We also provide a diagrammatic realization
of ${\mathsf {BC}}_{r,s}(q)$ as the $(r,s)$-bead tangle algebra ${\mathsf
{BT}}_{r,s}(q)$. Finally, we define the notion of $q$-Schur superalgebras of
type $\mathsf{Q}$ and establish their basic properties.
