If $Y_{i}$ is independently distributed as $N(\mu_i, \sigma^2)$, $i=1,\ldots,n$ and suppose $\mu_i=\beta_0 + \log(\beta_1+\beta_2 x_i)$. Is this a GLM?
So my train of thought was, if I separate it into separate cases of $\beta_0$ being known vs. unknown. But after that, I wouldn't even know how to begin.
If $\beta_0$ is known, then just linearize the model, i.e., $$ \exp\{\mu_i - \beta_0 \} = \beta_1 + \beta_2 x_i. $$ If $\beta_0$ is unknown, then you can perform the following linearization \begin{align} \exp\{\mu_i\} &= \exp\{\beta_0\}\exp\{\ln(\beta_1 + \beta_2 x_i\}\\ & = \beta_1\exp\{\beta_0\} + \exp\{\beta_0\}\beta_2x_i \\ & = b_0+b_1x_i, \end{align} however you won't be able to obtain the estimators of $\beta_1$, $\beta_2$ and $\beta_0$. Hence, it is a good approach in a case that you are interested in prediction rather than inference.
Another approach, as your model is non-linear, is to use non-linear regression. In this case you will have to to provide some initial value for $\beta_0$ (initial values for $\beta_1$ and $\beta_2$ can be obtained from a linear regression of the first aforementioned model once you'l provide some value for $\beta_0$).