Target tracking is one of the essential applications of the wireless sensor network. This paper considers the scenario where the target motion trajectory satisfies a quadratic equality constraint. In practice, when an aircraft hovers or a ground vehicle travels along a curve, its trajectory can be represented approximately as a quadratic function. This study applies quadratic constraints in the well-known Kalman filter (KF) to improve its performance in target tracking. The proposed algorithm first utilizes newly obtained positioning measurements and the unconstrained KF to produce an updated state estimation and then refines it using a maximum likelihood estimator (MLE) with quadratic equality constraints. When solving the constrained MLE problem, this paper formulates it as a generalized trust region sub-problem (GTRS) in order to obtain its globally optimal solution. Simulation results show that the proposed algorithm outperforms previously developed nonlinear KF algorithms with quadratic equality constraints in terms of enhanced target tracking accuracy.