As Francois Ziegler pointed out, it is not true in general. The map $\sigma: N \to gNg^{-1}$, $\sigma(n)=gng^{-1}$, is an isometry if and only if $$ \langle Ad(g)\cdot X,Ad(g)\cdot Y\rangle=\langle X,Y\rangle \quad\text{for all $X,Y \in\mathfrak n:=\textrm{Lie}(N)$.} $$ This follows since a left-invariant metric on a Lie group is determined by the inner product on the tangent space at the identity (identified with the Lie algebra), and also because the Lie algebra of $gNg^{-1}$ is $Ad(g)\cdot \mathfrak n$.
Note that a bi-invariant metric satisfies this condition, but the space of them may be usually larger. I am not sure whether there might be an example of an isometry between $N$ and $gNg^{-1}$ different than $\sigma$.