In order to protect superconducting magnets, quenches must be detected on time. Unfortunately, conventional simulation predictions are not accurate enough, because they often overestimate the quench detection thresholds. These false triggers lead to frequent and unneccessary shutdowns, which considerably reduce the availability of the entire system and become even more critical for the newly developed magnets based on the Nb$_3$Sn technology. While multiphysical three-dimensional (3D) simulations are computationally expensive, two-dimensional (2D) simulations of the magnet's cross-section lack accuracy. To increase quench prediction quality while keeping the computational effort low, quasi-3D simulation methods can be employed. Here, the cross-section of the magnet is discretized with linear finite elements, while the transversal direction is resolved with polynomial spectral elements. This hybrid approach achieves significantly smaller system of equations than in the standard 3D approach and provides a higher accuracy than in the 2D case. This work deals with the transient simulation of the heat propagation in a dipole magnet. Different spectral basis functions are investigated and the method is validated against a benchmark model.