A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter(GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF,two extensions are developed, namely square root generalized cubature quadrature Kalman filter(SR-GCQKF) and iterated generalized cubature quadrature Kalman filter(I-GCQKF). In SR-GCQKF,the QR decomposition is exploited to alter the Cholesky decomposition and both predicted and filtered error covariances have been propagated in square root format to make sure the numerical stability. In I-GCQKF, the measurement update step is executed iteratively to make full use of the latest measurement and a new terminal criterion is adopted to guarantee the increase of likelihood. Detailed numerical experiments demonstrate the superior performance on both tracking stability and estimation accuracy of I-GCQKF and SR-GCQKF compared with GCQKF. A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter (GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF, two extensions are developed, respectively square root generalized cubature quadrature Kalman filter (SR-GCQKF) and iterated generalized cubature quadrature In SR-GCQKF, the QR decomposition is exploited to alter the Cholesky decomposition and both predicted and filtered error covariances have been propagated in square root format to make sure the numerical stability. In I-GCQKF, the complete numerical use of the latest measurement and a new terminal criterion is taken to guarantee the increase of likelihood. make detailed use demonstrates the superior measurement of I-GCQKF and SR-GCQKF compared with GCQKF.