In the early 70s, Helton and Howe proved a beautiful formula for the trace of commutators of Toeplitz operators. In the 80s, Connes greatly generalized the Helton-Howe trace formula using cyclic cohomology. And the Connes-Chern character contains the Helton-Howe trace as the top degree component. In this talk, we will study the Connes-Chern character for the Toeplitz extension from the viewpoint of quantization by extending the Helton-Howe trace computation to semicommutators. As an outcome, we will establish the Helton-Howe trace formula for Toeplitz operators with $C^2$-symbols for all weighted Bergman spaces. This talk is based on joint work with Yi Wang and Dechao Zheng.