Natural G-quartets, a cyclic and coplanar array of four guanine residues held together through a Watson-Crick/Hoogsteen hydrogen-bond network, have received recently much attention due to their involvement in G-quadruplex DNA, an alternative higher-order DNA structure strongly suspected to play important roles in key cellular events. Besides this, synthetic G-quartets (SQ), which artificially mimic native G-quartets, have also been widely studied for their involvement in nanotechnological applications (i.e., nanowires, artificial ion channels, etc.). In contrast, intramolecular synthetic G-quartets (iSQ), also named template-assembled synthetic G-quartets (TASQ), have been more sparingly investigated, despite a technological potential just as interesting. Herein, we report on a particular iSQ named (PNA) DOTASQ, which demonstrates very interesting properties in terms of DNA and RNA interaction (notably its selective recognition of quadruplexes according to a bioinspired process) and catalytic activities, through its ability to perform peroxidase-like hemin-mediated oxidations either in an autonomous fashion (i.e., as pre-catalyst for TASQzyme reactions) or in conjunction with quadruplex DNA (i.e., as enhancing agents for DNAzyme processes). These results provide a solid scientific basis for TASQ to be used as multitasking tools for bionanotechnological applications.