Odd-frequency superconductivity is an exotic superconducting state in which the symmetry of the gap function is odd in frequency. Here, we show that an inherent odd-frequency mode emerges dynamically under application of a Lorentz transformation of the anomalous Green function with the general frequency-dependent gap function. To see this, we consider a Dirac model with quartic potential and perform a mean-field analysis to obtain a relativistic Bogoliubov-de Gennes system. Solving the resulting Gor'kov equations yields expressions for relativistic normal and anomalous Green functions. The form of the relativistically invariant pairing term is chosen such that it reduces to BCS form in the nonrelativistic limit. We choose an ansatz for the gap function in a particular frame which is even frequency and analyze the effects on the anomalous Green function under a boost into a relativistic frame. The odd-frequency pairing emerges dynamically as a result of the boost. In the boosted frame, the order parameter contains terms which are both even and odd in frequency. The relativistic correction to the anomalous Green function to first order in the boost parameter is completely odd in frequency. In this paper, we provide evidence that odd-frequency pairing may form intrinsically within relativistic superconductors.