The sigma-point filters, such as the unscented Kalman filter, are popular alternatives to the ubiquitous extended Kalman filter. The classical quadrature rules used in the sigma-point filters are motivated via polynomial approximation of the integrand; however, in the applied context, these assumptions cannot always be justified. As a result, a quadrature error can introduce bias into estimated moments, for which there is no compensatory mechanism in the classical sigma-point filters. This can lead in turn to estimates and predictions that are poorly calibrated. In this article, we investigate the Bayes-Sard quadrature method in the context of sigma-point filters, which enables uncertainty due to quadrature error to be formalized within a probabilistic model. Our first contribution is to derive the well-known classical quadratures as special cases of the Bayes-Sard quadrature method. Based on this, a general-purpose moment transform is developed and utilized in the design of a novel sigma-point filter, which explicitly accounts for the additional uncertainty due to quadrature error.