The main line of research in cancer treatment is the development of methods for early diagnosis and targeted drug delivery to cancer cells. Fluorescent semiconductor core/shell nanocrystals of quantum dots (e.g., CdSe/ZnS) conjugated with an anticancer drug, e.g., an acridine derivative, allow real-time tracking and control of the process of the drug delivery to tumors. However, linking of acridine derivatives to a quantum dot can be accompanied by quantum dot fluorescence quenching caused by electron transfer from the quantum dot to the organic molecule. In this work, it has been shown that the structure of the shell of the quantum dot plays the decisive role in the process of photoinduced charge transfer from the quantum dot to the acridine ligand, which is responsible for fluorescence quenching. It has been shown that multicomponent ZnS/CdS/ZnS shells of CdSe cores of quantum dots, which have a relatively small thickness, make it possible to significantly suppress a decrease in the quantum yield of fluorescence of quantum dots as compared to both the classical ZnS thin shell and superthick shells of the same composition. Thus, core/multicomponent shell CdSe/ZnS/CdS/ZnS quantum dots can be used as optimal fluorescent probes for the development of systems for diagnosis and treatment of cancer with the use of anticancer compounds based on acridine derivatives.