Given the joint density\begin{equation}
f(x, y) = \left[(1 - \theta)\,\alpha\,\beta + \theta\,\alpha^2\,\beta\,x + \theta\,\alpha\,\beta^2\,y +
\theta^2\,\alpha^2\,\beta^2\,x\,y\right]\,S(x, y)
\end{equation}
the conditional density of $Y$ given $X=x$ is such that
\begin{align}
f(y|x) &\propto \left[(1 - \theta)\,\alpha\,\beta + \theta\,\alpha^2\,\beta\,x\right]\, S(x, y)+ \left[\theta\,\alpha\,\beta^2 +
\theta^2\,\alpha^2\,\beta^2\,x\right]\,y\,S(x, y)\\
&\propto \left[1 - \theta+ \theta\,\alpha x\right]\, S(x, y)+ \left[\theta + \theta^2\,\alpha\,\beta\,x\right]\,y\,S(x, y)\\
&\propto \left[1 - \theta+ \theta\,\alpha x\right]\, \exp\left[- \left(\beta\,y + \theta\,\alpha\,\beta\,x\,y\right)\right]+ \left[\theta + \theta^2\,\alpha\,\beta\,x\right]\,y\,\exp\left[- \left( \beta\,y + \theta\,\alpha\,\beta\,x\,y\right)\right]\\
&\propto \left[1 - \theta+ \theta\,\alpha x\right]\, \exp\left[- \beta\,y\left( 1+ \theta\,\alpha\,x\right)\right]+ \left[\theta + \theta^2\,\alpha\,\beta\,x\right]\,y\,\exp\left[- \beta\,y\left( 1 + \theta\,\alpha x \right)\right]\\\end{align}
which means that the conditional distribution of $Y$ given $X=x$ is a mixture of two distributions,
- the exponential $\mathcal{E}(\beta\,\left\{ 1+ \theta\,\alpha\,x\right\})$ distribution
- the Gamma $\mathcal{G}a(2,\beta\,\left\{ 1+ \theta\,\alpha\,x\right\})$ distribution
with respective weights $1$ and $$ \beta^{-1}\left[\theta + \theta^2\,\alpha\,\beta\,x\right]\times\left( 1 + \theta\,\alpha x \right)^{-2}$$up to normalisation. It is therefore straightforward to simulate $X$ marginally and then $Y$ given this realisation of $X$.