Quadruplexes (GQs), peculiar DNA/RNA motifs concentrated in specific genomic regions, play a vital role in biological processes including telomere stability and, hence, represent promising targets for anticancer therapy. GQs are formed by folding guanine-rich sequences into square planar G-tetrads which stack onto one another. Metal cations, most often potassium, further stabilize the architecture by coordinating the lone electron pairs of the O atoms. The presence of additional nucleic acid bases, however, has been recently observed experimentally and contributes substantially to the structural heterogeneity of quadruplexes. Therefore, it is of paramount significance to understand the factors governing the underlying complex processes in these structures. The current study employs DFT calculations to model the interactions between metal cations (K+, Na+, Sr2+) and diverse tetrads composed of a guanine layer in combination with a guanine (G)-, adenine (A)-, cytosine (C)-, thymine (T)-, or uracil (U)-based tetrad layer. Moreover, the addition of 4-(3,4-dihydroisoquinolin-2-yl)-2-(quinolin-2-yl)quinazoline to the modeled quadruplexes as a possible mechanism of its well-exerted antitumor effect is assessed. The calculations imply that the metal cation competition and ligand complexation are influenced by the balance between electronic and implicit/explicit solvation effects, the composition of the tetrad layers, as well as by the solvent exposure to the surrounding environment expressed in terms of different dielectric constant values. The provided results significantly enhance our understanding of quadruplex diversity, ligand recognition, and the underlying mechanisms of stabilization at an atomic level.