Plasmon-exciton coupling is of great importance to many optical devices and applications. One of the coupling manifestations is plasmon-enhanced fluorescence. Although this effect is demonstrated in numerous experimental and theoretical works, there are different particle shapes for which this effect is not fully investigated. In this work electrostatic complexes of gold nanorods and CdSe/CdZnS quantum dots were studied. Double-resonant gold nanorods have an advantage of the simultaneous enhancement of the absorption and emission when the plasmon bands match the excitation and fluorescence wavelengths of an emitter. A relationship between the concentration of quantum dots in the complexes and the enhancement factor was established. It was demonstrated that the enhancement factor is inversely proportional to the concentration of quantum dots. The maximal fluorescence enhancement by 10.8 times was observed in the complex with the smallest relative concentration of 2.5 quantum dots per rod and approximately 5 nm distance between them. Moreover, the influence of quantum dot location on the gold nanorod surface plays an important role. Theoretical study and experimental data indicate that only the position near the nanorod ends provides the enhancement. At the same time, the localization of quantum dots on the sides of the nanorods leads to the fluorescence quenching.