  The first year of data from the Dark Energy Spectroscopic Instrument (DESI)
contains the largest set of Lyman-$\alpha$ (Ly$\alpha$) forest spectra ever
observed. This data, collected in the DESI Data Release 1 (DR1) sample, has
been used to measure the Baryon Acoustic Oscillation (BAO) feature at redshift
$z=2.33$. In this work, we use a set of 150 synthetic realizations of DESI DR1
to validate the DESI 2024 Ly$\alpha$ forest BAO measurement. The synthetic data
sets are based on Gaussian random fields using the log-normal approximation. We
produce realistic synthetic DESI spectra that include all major contaminants
affecting the Ly$\alpha$ forest. The synthetic data sets span a redshift range
$1.8<z<3.8$, and are analysed using the same framework and pipeline used for
the DESI 2024 Ly$\alpha$ forest BAO measurement. To measure BAO, we use both
the Ly$\alpha$ auto-correlation and its cross-correlation with quasar
positions. We use the mean of correlation functions from the set of DESI DR1
realizations to show that our model is able to recover unbiased measurements of
the BAO position. We also fit each mock individually and study the population
of BAO fits in order to validate BAO uncertainties and test our method for
estimating the covariance matrix of the Ly$\alpha$ forest correlation
functions. Finally, we discuss the implications of our results and identify the
needs for the next generation of Ly$\alpha$ forest synthetic data sets, with
the top priority being to simulate the effect of BAO broadening due to
non-linear evolution.
