The paper investigates the shock cone instabilities and quasi-periodic oscillations (QPOs) around black holes, focusing on Hartle–Thorne black holes. The authors numerically model the dynamical and oscillation properties of shock cones formed around both slowly and rapidly rotating Hartle–Thorne black holes, driven by the Bondi–Hoyle–Lyttleton (BHL) accretion mechanism. They find that an increase in the quadrupole parameter leads to a decrease in the shock cone opening angle, with more matter falling into the black hole within the cone. The combination of the quadrupole parameter and black hole rotation causes chaotic motion inside the cone, exciting fundamental oscillation modes and forming new frequencies due to nonlinear coupling. The study also compares the outcomes with those from Kerr and Einstein–Gauss–Bonnet (EGB) gravities, highlighting the impact of the quadrupole parameter on shock cones and QPOs. The results provide insights into the dynamics of black hole systems and the physics of accretion disks, contributing to the understanding of X-ray emissions and observational data from black hole binaries.The paper investigates the shock cone instabilities and quasi-periodic oscillations (QPOs) around black holes, focusing on Hartle–Thorne black holes. The authors numerically model the dynamical and oscillation properties of shock cones formed around both slowly and rapidly rotating Hartle–Thorne black holes, driven by the Bondi–Hoyle–Lyttleton (BHL) accretion mechanism. They find that an increase in the quadrupole parameter leads to a decrease in the shock cone opening angle, with more matter falling into the black hole within the cone. The combination of the quadrupole parameter and black hole rotation causes chaotic motion inside the cone, exciting fundamental oscillation modes and forming new frequencies due to nonlinear coupling. The study also compares the outcomes with those from Kerr and Einstein–Gauss–Bonnet (EGB) gravities, highlighting the impact of the quadrupole parameter on shock cones and QPOs. The results provide insights into the dynamics of black hole systems and the physics of accretion disks, contributing to the understanding of X-ray emissions and observational data from black hole binaries.