Strangely I haven't found a derivation of the E.O.M. of a string in an arbitrary background metric $G_{\mu \nu}$ from the Nambu-Goto action. (Many places present it for the Polyakov action.) Upon attempting to derive it for myself, however, I find there is an extra term that shouldn't be there.
The Nambu-Goto action is $$ S_{NG} = -T \int d^2 \sigma \sqrt{-h} $$ where $$ h_{ab} := G_{\alpha \beta}(X) \partial_a X^\alpha \partial_b X^\beta. $$ The Euler-Lagrange equation is $$ 0 = \frac{\partial \mathcal{L}}{ \partial X^\mu} - \partial_c \frac{\partial \mathcal{L}}{ \partial \partial_c X^\mu}. $$ Using $$ \delta \sqrt{-h} = \frac{1}{2} \sqrt{-h} h^{ab} \delta h_{ab} $$ we get $$ \frac{\partial \mathcal{L}}{ \partial X^\mu} = -\frac{T}{2}\sqrt{-h} h^{ab} (\partial_\mu G_{\alpha \beta}) \partial_a X^\alpha \partial_b X^\beta $$ for the first term in the Euler-Lagrange equation. For the second term, we get $$ \partial_c \frac{\partial \mathcal{L}}{ \partial \partial_c X^\mu} = -T \partial_c(\sqrt{-h} h^{ac} G_{\alpha \mu} \partial_a X^\alpha). $$ Here we recognize the $\partial_c \frac{\partial \mathcal{L}}{ \partial \partial_c X^\mu}$ term as the rightful equation of motion, derivable from the Polyakov action. However, the other term, $\frac{\partial \mathcal{L}}{ \partial X^\mu}$, is puzzling me. Shouldn't it be $0$?
