  We present the one-dimensional Lyman-$\alpha$ forest power spectrum
measurement using the first data provided by the Dark Energy Spectroscopic
Instrument (DESI). The data sample comprises $26,330$ quasar spectra, at
redshift $z > 2.1$, contained in the DESI Early Data Release and the first two
months of the main survey. We employ a Fast Fourier Transform (FFT) estimator
and compare the resulting power spectrum to an alternative likelihood-based
method in a companion paper. We investigate methodological and instrumental
contaminants associated to the new DESI instrument, applying techniques similar
to previous Sloan Digital Sky Survey (SDSS) measurements. We use synthetic data
based on log-normal approximation to validate and correct our measurement. We
compare our resulting power spectrum with previous SDSS and high-resolution
measurements. With relatively small number statistics, we successfully perform
the FFT measurement, which is already competitive in terms of the scale range.
At the end of the DESI survey, we expect a five times larger Lyman-$\alpha$
forest sample than SDSS, providing an unprecedented precise one-dimensional
power spectrum measurement.
