The time-dependent mode structure of the Lyapunov vectors associated with the stepwise structure of the Lyapunov spectra and its relation to the momentum autocorrelation function are discussed in quasi-one-dimensional many-hard-disk systems. We obtain the complete mode structures (Lyapunov modes) for all components of the Lyapunov vectors, including the longitudinal and transverse components of both the spatial and momentum parts, and their phase relations. These mode structures are suggested by the form of the Lyapunov vectors for the zero-Lyapunov exponents. The spatial node structures of these modes are explained by the reflection properties of the hard walls used in the models. Our main result is that the largest time-oscillating period of the Lyapunov modes is twice as long as the time-oscillating period of the longitudinal momentum autocorrelation function. This relation is satisfied irrespective of the number of particles and the boundary conditions. A simple explanation for this relation is given based on the form of the time-dependent Lyapunov mode.