Over-smoothing (OSM) and over-squashing (OSQ) are two fundamental phenomena that limit the performance of Graph Neural Networks (GNNs), yet a unified spectral-spatial understanding of these phenomena remains underexplored. In this paper, we adopt polynomial spectral filters as an analytical tool to establish a unified spectral-spatial framework for graph convolution and systematically characterize the effect of the polynomial order k on information propagation in GNNs. Within this framework, we reveal an intrinsic trade-off induced by the polynomial order. Specifically, higher-order filters enhance spectral expressiveness and alleviate OSM caused by the dominant low-frequency components. However, they also expand the spatial receptive field, thereby intensifying information compression and increasing the risk of OSQ. Based on this analysis, we provide a principled guideline for selecting the polynomial order and propose a Quadratic Spectral Graph Convolution Network (QS-GCN) for graph classification. Experiments demonstrate the effectiveness and robustness of the proposed method.