Streamflows often vary strongly with season, and this leads to seasonal dependence in hydrological model errors and prediction uncertainty. In this study, we introduce three error models to describe errors from a monthly rainfall-runoff model: a seasonally invariant model, a seasonally variant model, and a hierarchical error model. The seasonally variant model and the hierarchical error model use month-specific parameters to explicitly account for seasonal dependence, while the seasonally invariant model does not. A Bayesian prior is used in the hierarchical error model to account for potential variation and connection among model parameters of different months. The three error models are applied to predicting streamflows for five Australian catchments and are compared by various performance scores and diagnostic plots. The seasonally variant model and the hierarchical model both perform substantially better than the seasonally invariant model. From a cross-validation analysis, the hierarchical error model provides both the most accurate prediction mean and the most reliable prediction uncertainty distribution in most situations. The use of the prior to constrain the model parameters in the hierarchical model produces more robust parameter estimation than the other two models.