This paper proposes and analyzes a magnon-skyrmion hybrid quantum system, consisting of a micromagnet and nearby magnetic skyrmions. The study predicts a strong coupling mechanism between the magnonic mode of the micromagnet and the quantized helicity degree of freedom of the skyrmion. The hybrid system enables the induction of magnon-mediated nonreciprocal interactions and responses between distant skyrmion qubits or between skyrmion qubits and other quantum systems like superconducting qubits. This work provides a quantum platform for investigating diverse quantum effects and quantum information processing with magnetic microstructures. The system is composed of a YIG micromagnet and a skyrmion, where the skyrmion is located beneath the micromagnet. The micromagnet can adopt various shapes such as spheres, square or circular dots. The skyrmion's spin direction is given by a normalized spin vector, with the helicity being an internal degree of freedom. The skyrmion qubit can be constructed by quantizing the helicity. The system's Hamiltonian is derived, and the interaction between the magnon and the skyrmion is analyzed. The coupling strength is calculated, and the system is shown to reach the strong-coupling regime. The study also explores the exponential enhancement of coupling strength through parametric amplification techniques, which allows for stronger coupling between skyrmion qubits and larger YIG spheres. Nonreciprocal interactions between skyrmion qubits and superconducting qubits are also investigated, demonstrating the potential for nonreciprocal population conversion between qubits. The experimental feasibility of the proposed system is discussed, with parameters such as the effective spin, lattice spacing, and anisotropy energy provided. The system's resonant frequency and coupling strength are calculated, and the hybrid quantum system's cooperativity is analyzed. The study concludes that the proposed system can achieve strong coupling and enable quantum information processing with magnetic microstructures.This paper proposes and analyzes a magnon-skyrmion hybrid quantum system, consisting of a micromagnet and nearby magnetic skyrmions. The study predicts a strong coupling mechanism between the magnonic mode of the micromagnet and the quantized helicity degree of freedom of the skyrmion. The hybrid system enables the induction of magnon-mediated nonreciprocal interactions and responses between distant skyrmion qubits or between skyrmion qubits and other quantum systems like superconducting qubits. This work provides a quantum platform for investigating diverse quantum effects and quantum information processing with magnetic microstructures. The system is composed of a YIG micromagnet and a skyrmion, where the skyrmion is located beneath the micromagnet. The micromagnet can adopt various shapes such as spheres, square or circular dots. The skyrmion's spin direction is given by a normalized spin vector, with the helicity being an internal degree of freedom. The skyrmion qubit can be constructed by quantizing the helicity. The system's Hamiltonian is derived, and the interaction between the magnon and the skyrmion is analyzed. The coupling strength is calculated, and the system is shown to reach the strong-coupling regime. The study also explores the exponential enhancement of coupling strength through parametric amplification techniques, which allows for stronger coupling between skyrmion qubits and larger YIG spheres. Nonreciprocal interactions between skyrmion qubits and superconducting qubits are also investigated, demonstrating the potential for nonreciprocal population conversion between qubits. The experimental feasibility of the proposed system is discussed, with parameters such as the effective spin, lattice spacing, and anisotropy energy provided. The system's resonant frequency and coupling strength are calculated, and the hybrid quantum system's cooperativity is analyzed. The study concludes that the proposed system can achieve strong coupling and enable quantum information processing with magnetic microstructures.