Tensors don't change under the transformations they are tensors for. That's a definition. What changes in general are the components of tensors with respect to basis transformations.
Denoting the Minkowski product of two four vectors with ##\boldsymbol{x} \cdot \boldsymbol{y}## the metric (or better pseudometric!) tensor's components with respect to an arbitrary basis ##\boldsymbol{b}_{\mu}## are given by
$$g_{\mu \nu} = \boldsymbol{b}_{\mu} \cdot \boldsymbol{b}_{\nu}.$$
A Lorentz transformation by definition is a linear transformation which leaves the Minkowski products between any two vectors invariant. So defining a new basis via a Lorentz transformation ##\boldsymbol{b}_{\mu}'=\Lambda \boldsymbol{b}_{\mu}## implies
$$g_{\mu \nu}'= \boldsymbol{b}_{\mu}' \cdot \boldsymbol{b}_{\nu}' = (\Lambda \boldsymbol{b}_{\mu}) \cdot (\Lambda \boldsymbol{b}_{\nu}) = \boldsymbol{b}_{\mu} \cdot \boldsymbol{b}_{\nu}=g_{\mu \nu},$$
i.e., if you change a basis by using Lorentz transformations, the components of the pseudometric don't change.
This is particularly true for pseudoorthonormal bases, for which
$$g_{\mu \nu}=\eta_{\mu \nu} =\mathrm{diag}(1,-1,-1,-1).$$