In your case, you are putting a Gaussian prior on erro_y, which I will denote \epsilon=g(y), with g(\cdot) being the transformation. The change of variables formula says that we can find the equivalent distribution over y as p(y)=|J_{g}|p_{\epsilon}(g(y))
\epsilon=g(y)=\frac{y-\beta_0+\beta_1x}{1+\beta_1^2}\Rightarrow J_g=\frac{1}{1+\beta_1^2}
As this is a constant you can safely ignore it as Stan’s target only needs to be accurate up to a constant.
For the record, if you put priors on your betas instead, you won’t need the Jacobian term in general.