This article investigates the shock cone instabilities and quasi-periodic oscillations (QPOs) around Hartle–Thorne black holes, using the Bondi–Hoyle–Lyttleton (BHL) accretion mechanism. Numerical simulations show that increasing the quadrupole parameter reduces the shock cone opening angle and decreases the amount of matter falling into the black hole. The combination of the quadrupole parameter and black hole rotation leads to chaotic motion within the cone, exciting fundamental oscillation modes and generating new frequencies through nonlinear coupling. The study compares results from Hartle–Thorne gravity with those from Kerr and Einstein–Gauss–Bonnet (EGB) gravities, highlighting the impact of the quadrupole parameter on shock cones and QPOs. The Hartle–Thorne metric describes spacetime around slowly rotating and slightly deformed compact objects, incorporating the quadrupole parameter. It extends the Schwarzschild metric by including rotation effects and is more general than the Kerr solution. The study uses this metric to model shock cones around both slowly and rapidly rotating black holes, analyzing how the quadrupole parameter and black hole rotation affect the cone's structure and QPO frequencies. For slowly rotating black holes (a/M = 0.4), the shock cone opening angle decreases with increasing quadrupole parameter (q), leading to changes in QPO frequencies. The shock cone's dynamics, including density, velocity, and pressure variations, are analyzed, revealing how the quadrupole parameter influences the cone's behavior and the resulting QPOs. The power spectrum density (PSD) analysis shows that QPOs are persistent and follow 1:2:3... ratios, with higher observability at higher q values. For rapidly rotating black holes (a/M = 0.9), the shock cone opening angle is larger, and the cone exhibits strong instability even after reaching a steady state. The PSD analysis reveals significant QPO frequencies and nonlinear couplings, indicating that the quadrupole parameter and black hole rotation significantly affect the cone's dynamics and QPO characteristics. The study highlights the importance of the Hartle–Thorne metric in understanding the complex interplay between black hole rotation, quadrupole parameters, and QPOs in astrophysical systems.This article investigates the shock cone instabilities and quasi-periodic oscillations (QPOs) around Hartle–Thorne black holes, using the Bondi–Hoyle–Lyttleton (BHL) accretion mechanism. Numerical simulations show that increasing the quadrupole parameter reduces the shock cone opening angle and decreases the amount of matter falling into the black hole. The combination of the quadrupole parameter and black hole rotation leads to chaotic motion within the cone, exciting fundamental oscillation modes and generating new frequencies through nonlinear coupling. The study compares results from Hartle–Thorne gravity with those from Kerr and Einstein–Gauss–Bonnet (EGB) gravities, highlighting the impact of the quadrupole parameter on shock cones and QPOs. The Hartle–Thorne metric describes spacetime around slowly rotating and slightly deformed compact objects, incorporating the quadrupole parameter. It extends the Schwarzschild metric by including rotation effects and is more general than the Kerr solution. The study uses this metric to model shock cones around both slowly and rapidly rotating black holes, analyzing how the quadrupole parameter and black hole rotation affect the cone's structure and QPO frequencies. For slowly rotating black holes (a/M = 0.4), the shock cone opening angle decreases with increasing quadrupole parameter (q), leading to changes in QPO frequencies. The shock cone's dynamics, including density, velocity, and pressure variations, are analyzed, revealing how the quadrupole parameter influences the cone's behavior and the resulting QPOs. The power spectrum density (PSD) analysis shows that QPOs are persistent and follow 1:2:3... ratios, with higher observability at higher q values. For rapidly rotating black holes (a/M = 0.9), the shock cone opening angle is larger, and the cone exhibits strong instability even after reaching a steady state. The PSD analysis reveals significant QPO frequencies and nonlinear couplings, indicating that the quadrupole parameter and black hole rotation significantly affect the cone's dynamics and QPO characteristics. The study highlights the importance of the Hartle–Thorne metric in understanding the complex interplay between black hole rotation, quadrupole parameters, and QPOs in astrophysical systems.