> I want to get an expression for $\nabla_{a}g_{bc}$

If you are using the Levi-Civita connection, then the covariant derivative of the metric is *zero* by the *definition* of that connection.

See [this PSE question](https://physics.stackexchange.com/questions/47919/why-is-the-covariant-derivative-of-the-metric-tensor-zero).