The posterior distribution for nonlinear Bayesian inverse problems often has to be estimated via sampling and requires many simulations of the forward model, which can be computationally expensive when the forward model requires simulating a high-dimensional dynamical system. This can be remedied by using a reduced forward model that captures the important dynamics of the high-dimensional dynamical system. In systems theory, balanced truncation methods obtain efficient reduced models by projecting the high-dimensional model operators onto the space spanned by dominant eigenvectors of the system Gramians. In this paper, we consider Bayesian smoothing problems for quadratic dynamical systems and introduce inference-oriented Gramians that define a procedure for model reduction by balanced truncation. We provide a stability analysis of the resulting quadratic nonlinear model and support the analysis with a numerical example.