= Solution
Use the random-walk proposal specified by its centre and <covariance>: for the current state $z=(x,y)$ draw $Z_1,Z_2$ independently standard normal and set
$$
x'=x+\sigma_qZ_1,\qquad y'=y+\sigma_qZ_2,\qquad\sigma_q>0.
$$
Its properly normalized density of the <proposal distribution> is
$$
q(z'\mid z)=\frac1{2\pi\sigma_q^2}\exp\left(-\frac{(x'-x)^2+(y'-y)^2}{2\sigma_q^2}\right).
$$
It is symmetric: $q(z'\mid z)=q(z\mid z')$. This is a centred-at-the-current-state proposal, rather than an unshifted draw from the density notation in the source. The <Random-walk Metropolis algorithm> consequently needs only the target ratio. Put $H(x,y)=x^2+y^2-2\rho xy$. Then
$$
\boxed{\alpha(z,z')=\min\left\{1,\exp\left(-\frac{H(x',y')-H(x,y)}{2(1-\rho^2)}\right)\right\}.}
$$
An <independent> uniform decides the move; on rejection both coordinates remain at their previous values. A proposal with lower $H$ is always accepted. This construction uses the joint density directly, without needing either <conditional distribution>. Its continuous positive proposal density supplies communication over the target support, and the holding probability avoids a deterministic periodic transition pattern.
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