The hybrid integration of two-dimensional (2D) materials on various photonic integration platforms has attracted widespread research interest because of the new functionalities enabled by the 2D materials for applications in photodetection, optical modulation and nonlinear optical signal processing. Tellurium is known to have high mobility, and quasi-2D tellurium is stable in air and has a small bandgap that may make it suitable for platform-independent scalable integration of high-performance photodetectors in the infrared band. In this work, we propose and implement a new structure for integrating tellurium with silicon nitride (SiN) waveguides, adding photodetector capability to an otherwise passive waveguide platform. At a wavelength of 1570 nm, the fabricated tellurium photodetector has an experimentally measured responsivity of 0.5 A W-1 at 1 V bias voltage and a bandwidth of 12 GHz. We show that the bandwidth is not limited by the carrier transit time, but rather by the RC time constant, which can be further improved.