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In the era of Internet of Things (IoT), multi-access edge computing (MEC)-enabled satellite-aerial-terrestrial integrated network (SATIN) has emerged as a promising technology to provide massive IoT devices with seamless and reliable communication and computation services. This paper investigates the cooperation of low Earth orbit (LEO) satellites, high altitude platforms (HAPs), and terrestrial base stations (BSs) to provide relaying and computation services for vastly distributed IoT devices. Considering the uncertainty in dynamic SATIN systems, we formulate a stochastic optimization problem to minimize the time-average expected service delay by jointly optimizing resource allocation and task offloading while satisfying the energy constraints. To solve the formulated problem, we first develop a Lyapunov-based online control algorithm to decompose it into multiple one-slot problems. Since each one-slot problem is a large-scale mixed-integer nonlinear program (MINLP) that is intractable for classical computers, we further propose novel hybrid quantum-classical generalized Benders' decomposition (HQCGBD) algorithms to solve the problem efficiently by leveraging quantum advantages in parallel computing. Numerical results validate the effectiveness of the proposed MEC-enabled SATIN schemes.