A quick-hardening track (QHT) was developed by injecting quick-hardening mortar into an existing ballast track to rapidly substitute the ballast track with a slab track, thereby improving maintainability and running safety. QHT tracks on a bridge undergo track–bridge interactions similar to other track systems. This paper presents a model to analyze the interaction between the QHT and the bridge. This model considers the longitudinal resistances of rail fasteners and anchors, as well as the interlayer friction between the track and the bridge. A sequential analysis method was applied to systematically consider such effects, revealing that rail additional stress will be high if the track slips over the bridge for a very low frictional coefficient of 0.1. Furthermore, a track segment without an anchor can slip under train traction load when the frictional coefficient is 0.3 or lower. For low friction cases, low-speed operation is advised to prevent the accumulation of the resulting longitudinal slip displacements of the track. An anchor should be installed immediately after the quick-hardening mortar provides sufficient bearing strength to the anchors. The proposed sequential analysis is useful for determining the critical friction coefficient and appropriate longitudinal resistance of a rail fastener, as well as for verifying track safety.