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The baryon acoustic oscillation (BAO) scale acts as a standard ruler for measuring cosmological distances and has therefore emerged as a leading probe of cosmic expansion history. However, any physical effect that alters the length of the ruler can lead to a bias in our determination of distance and expansion rate. One of these physical effects is the streaming velocity, the relative velocity between baryons and dark matter in the early Universe, which couples to the BAO scale due to their common origin in acoustic waves at recombination. In this work, we investigate the impact of streaming velocity on the BAO feature of the Lyman-$α$ forest auto-power spectrum, one of the main tracers being used by the recently commissioned DESI spectrograph. To do this, we develop a new perturbative model for Lyman-$α$ flux density contrast which is complete to second order for a certain set of fields, and applicable to any redshift-space tracer of structure since it is based only on symmetry considerations. We find that there are 8 biasing coefficients through second order. We find streaming velocity-induced shifts in the BAO scale of 0.081--0.149% (transverse direction) and 0.053--0.058% (radial direction), depending on the model for the biasing coefficients used. These are smaller than, but not negligible compared to, the DESI Lyman-$α$ BAO error budget, which is 0.46% on the overall scale. The sensitivity of these results to our choice of bias parameters underscores the need for future work to measure the higher-order biasing coefficients from simulations, especially for future experiments beyond DESI.