Quantum speed limit (QSL) in a quantum system originates from the essential principle of the quantum mechanics. It gives a maximum speed of evolution or a minimum evolution time of the quantum system, which has potential applications in the fields of quantum information, quantum communication and quantum control and so on. In the last decades, the QSL bounds have been explored from the isolated quantum systems to the open quantum systems, several different geometric measures have been adopted to investigate the distinguishability between the initial and the evolved state. The QSL bounds in many systems have been discussed, indicating that the tightness of the QSL metric is related to the researched dynamical system. However, the QSL problem for the molecular system has rarely been reported. The study of the quantum speed limits in the different molecular systems is helpful for realizing the quantum information technology based on the molecules. In this paper, the generalized quantum speed limit metric for linear molecular dynamics is studied by the algebraic approach. The quantum Fisher information metric and the Wigner-Yanase information metric are both used to study the QSLs in the dynamical evolution of the two linear molecules. Here the dynamical evolutions begin with the two kinds of vibrational states, Fock initial state and coherent initial state. The results show that the quantum Fisher information metric is more appropriate than the Wigner-Yanase information metric for HCN and DCN molecules. The relative differences between the generalized geometric QSL and the two geodesic QSL metrics become bigger gradually with the increase of the initial vibrational quantum number. However, the relative difference for the DCN molecule is smaller than for the HCN molecule. The relative difference between the strong coherent states is smaller, which indicates that the generalized geometric quantum speed limit is suitable to describing high-coherent case. In conclusion, the different QSL metrics reveal the discrepancy in the evolution of the molecular system, and the relative difference is related to the initial state of the molecules and molecular parameters. More molecular systems need to be investigated in order to obtain the criteria between the QSL metrics and the molecular parameters.