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Accurate modeling of active galactic nucleus (AGN) feedback, especially due to relativistic jets, is crucial for understanding the cool-core problem in galaxy clusters. We present a new subgrid method to model accretion onto and feedback from AGN in hydrodynamical simulations of galaxy clusters. Instead of applying the traditional Bondi formalism, we use a sink particle algorithm in which the accretion flux is measured directly through a control surface. A weighting kernel is used to reset the gas properties within the accretion radius at the end of each time step. We implement feedback in the form of bipolar jets whose properties are tied to the accretion rate. The method is tested with a spherically symmetric Bondi gas flow problem and a Bondi–Hoyle–Lyttleton wind problem, with and without jet feedback. We discuss the reliability of this model by comparing our jet simulations with those in the literature, and we examine the dependence of test results on parameters such as the resolution and size of the jet injection region. We find that the sink particle model can account for the α factor in accretion measurement, and the accretion radius must be resolved with at least two zones to produce realistic black hole accretion. We also show how underresolving the AGN feedback region in simulations can impact the feedback energy deposited and the jet dynamics. The code described here is the framework for a feedback model, described in a companion paper, that will use accretion disk modeling to more self-consistently determine the feedback efficiency.