We present an algorithm for detecting and tracking underwater mobile objects using active acoustic transmission of broadband chirp signals whose reflections are received by a hydrophone array. The method overcomes the problem of high false alarm rate by applying a track-before-detect approach to the sequence of received reflections. A 2D time-space matrix is created for the reverberations received from each transmitted probe signal by performing delay and sum beamforming and pulse compression. The result is filtered by a 2D constant false alarm rate (CFAR) detector to identify reflection patterns corresponding to potential targets. Closely spaced signals for multiple probe transmissions are combined into blobs to avoid multiple detections of a single object. A track-before-detect method using a Nearly Constant Velocity (NCV) model is employed to track multiple objects. The position and velocity is estimated by the debiased converted measurement Kalman filter. Results are analyzed for simulated scenarios and for experiments at sea, where GPS tagged gilt-head seabream fish were tracked. Compared to two benchmark schemes, the results show a favorable track continuity and accuracy that is robust to the choice of detection threshold.