We are concerned with the problem of hyperparameter selection for the fitted Q-evaluation (FQE). FQE is one of the state-of-the-art method for offline policy evaluation (OPE), which is essential to the reinforcement learning without environment simulators. However, like other OPE methods, FQE is not hyperparameter-free itself and that undermines the utility in real-life applications. We address this issue by proposing a framework of approximate hyperparameter selection (AHS) for FQE, which defines a notion of optimality (called selection criteria) in a quantitative and interpretable manner without hyperparameters. We then derive four AHS methods each of which has different characteristics such as distribution-mismatch tolerance and time complexity. We also confirm in experiments that the error bound given by the theory matches empirical observations.