Ternary quantum dots (QDs) are very promising nanomaterials with a range of potential applications in photovoltaics, light-emitting devices, and biomedicine. Despite quite intensive studies of ternary QDs over the last years, the specific relaxation channels involved in their emission mechanisms are still poorly understood, particularly in the corresponding core-shell nanostructures. In the present work, we have studied the recombination pathways of AgInS2 QDs stabilized with the ZnAgInS alloy layer and the ZnS shell (AIS/ZAIS/ZnS QDs) using time-resolved fluorescence spectroscopy. We have also investigated FRET in complexes of AIS/ZAIS/ZnS QDs and cyanine dyes with the absorption bands overlapping in the different regions of the QD emission spectrum, which allowed us to selectively quench the radiative transitions of the QDs. Our studies have demonstrated that FRET from QDs to dyes results in decreasing of all QD PL decay components with the shortest lifetime decreasing the most and the longest one decreasing the least. This research presents important approaches for the investigation of ternary QD luminescence mechanisms by the selective quenching of recombination pathways. These studies are also essential for potential applications of ternary QDs in photodynamic therapy, multiplex analysis, and time-resolved FRET sensing.