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The energy efficiency and output power of a quantum thermoelectric system with multiple electric currents and only one heat current are studied. The system is connected to the hot heat bath (cold bath) through one terminal (multiple terminals). In such configurations, there are multiple thermoelectric effects coexisting in the system. Using the Landauer-B(u)ttiker formalism, we show that the cooperation between the two thermoelectric effects in the three-terminal thermoelec-tric systems can lead to markedly improved performance of the heat engine. Such improvement also occurs in four-terminal thermoelectric heat engines with three output electric currents. Cooperative effects in these multi-terminal thermoelectric systems can considerably enlarge the physical parameter region that realizes high energy efficiency and output power. For refrigeration, we find that the energy efficiency can also be substantially improved by exploiting the cooperative effects in multi-terminal thermoelectric systems. All these results reveal a useful approach toward high-performance thermoelectric energy conversion in multi-terminal mesoscopic systems.