The linear elasticity, Stokes and Navier-Stokes examples are examples where systems of coupled PDEs are solved. Examples thus exist. The question is thus how to translate these examples to your needs.
My suggestion is thus to solve one PDE first to gain familiarity with gridap.
Next I would to consider two coupled PDEs, and sees how that goes. What might help is to gain a familiarity with the language (variational formulation and its spatial discretization) that gridap employs.
Does this make sense?