The fluctuation dissipation relation is usually stated in terms of an identity that relates the retarded, advanced and either the Keldysh or time-ordered correlators. This is easily enforced in Keldysh theory.
Considering a quantum system interacting with a bath in the Keldysh formalism, integrating out the bath, and taking the saddle points of the system action, we obtain dynamical equations that describe the evolution of an open system and automatically satisfy the fluctuation dissipation relation (see eg the first few chapters of Kamenev 2011, or his notes https://arxiv.org/abs/cond-mat/0412296)
This dynamical equation will generally not be the same as the dynamics obtained from the Lindblad equation.
Is there a good understanding of why this is? Presumably there are approximations in one approach that the other manages to avoid? Are the different regimes of applicability well understood?
Does the Lindblad equation also satisfy a fluctuation dissipation relation? (though I am not entirely sure the correct way to defined one in this context)
I can give more details if this question is unclear, but I have erred on the side of generality.