= Solution
For $q_\phi=N(\mu_Q,\sigma_Q^2)$,
$$
\operatorname{ELBO}(\mu_Q,\sigma_Q^2)
=-\frac12\log(2\pi\sigma^2)
-\frac{(y-\mu_Q)^2+\sigma_Q^2}{2\sigma^2}
-\frac12\left[
\log\frac{\tau^2}{\sigma_Q^2}
+\frac{\sigma_Q^2+\mu_Q^2}{\tau^2}-1
\right].
$$
Differentiation gives
$$
\mu_Q^*=\frac{\tau^2}{\sigma^2+\tau^2}y=\widetilde\theta,
\qquad
(\sigma_Q^2)^*=\frac{\sigma^2\tau^2}{\sigma^2+\tau^2}=\sigma_\theta^2.
$$
The Gaussian variational family contains the exact posterior, so its best member is the posterior itself and the maximized ELBO equals $\log Z$.
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