  Let $\mathbb F_q$ be a finite field, where $q$ is an odd prime power. Let
$R=\mathbb{F}_q+u\mathbb{F}_q+v\mathbb{F}_q+uv\mathbb F_q$ with
$u^2=u,v^2=v,uv=vu$. In this paper, we study the algebraic structure of
$(\theta, \Theta)$-cyclic codes of block length $(r,s )$ over $\mathbb{F}_qR.$
Specifically, we analyze the structure of these codes as left
$R[x:\Theta]$-submodules of $\mathfrak{R}_{r,s} =
\frac{\mathbb{F}_q[x:\theta]}{\langle x^r-1\rangle} \times
\frac{R[x:\Theta]}{\langle x^s-1\rangle}$. Our investigation involves
determining generator polynomials and minimal generating sets for this family
of codes. Further, we discuss the algebraic structure of separable codes. A
relationship between the generator polynomials of $(\theta, \Theta)$-cyclic
codes over $\mathbb F_qR$ and their duals is established. Moreover, we
calculate the generator polynomials of dual of $(\theta, \Theta)$-cyclic codes.
As an application of our study, we provide a construction of quantum
error-correcting codes (QECCs) from $(\theta, \Theta)$-cyclic codes of block
length $(r,s)$ over $\mathbb{F}_qR$. We support our theoretical results with
illustrative examples.
