Currently, effective drugs for triple-negative breast cancer (TNBC) are lacking in clinics. c-myc is one of the core members during TNBC tumorigenesis, and G-rich sequences in the promoter region can form a G-quadruplex conformation, indicating that the c-myc inhibitor is a possible strategy to fight cancer. Herein, a series of chiral ruthenium(II) complexes ([Ru(bpy)2(DPPZ-R)](ClO4)2, Λ/Δ-1: R = -H, Λ/Δ-2: R = -Br, Λ/Δ-3: R = -C≡C(C6H4)NH2) were researched based on their interaction with c-myc G-quadruplex DNA. Λ-3 and Δ-3 show high affinity and stability to decrease their replication. Additional studies showed that Λ-3 and Δ-3 exhibit higher inhibition against different tumor cells than other molecules. Δ-3 decreases the viability of MDA-MB-231 cells with an IC50 of 25.51 μM, which is comparable with that of cisplatin, with an IC50 of 25.9 μM. Moreover, Δ-3 exhibits acceptable cytotoxic activity against MDA-MB-231 cells in a zebrafish xenograft breast cancer model. Further studies suggested that Δ-3 decreases the viability of MDA-MB-231 cells predominantly through DNA-damage-mediated apoptosis, which may be because Δ-3 can induce DNA damage. In summary, the results indicate that Ru(II) complexes containing alkinyl groups can be developed as c-myc G-quadruplex DNA binders to block TNBC progression.