  The one-dimensional power spectrum $P_{\mathrm{1D}}$ of the Ly$\alpha$ forest
provides important information about cosmological and astrophysical parameters,
including constraints on warm dark matter models, the sum of the masses of the
three neutrino species, and the thermal state of the intergalactic medium. We
present the first measurement of $P_{\mathrm{1D}}$ with the quadratic maximum
likelihood estimator (QMLE) from the Dark Energy Spectroscopic Instrument
(DESI) survey early data sample. This early sample of $54~600$ quasars is
already comparable in size to the largest previous studies, and we conduct a
thorough investigation of numerous instrumental and analysis systematic errors
to evaluate their impact on DESI data with QMLE. We demonstrate the excellent
performance of the spectroscopic pipeline noise estimation and the impressive
accuracy of the spectrograph resolution matrix with two-dimensional image
simulations of raw DESI images that we processed with the DESI spectroscopic
pipeline. We also study metal line contamination and noise calibration
systematics with quasar spectra on the red side of the Ly$\alpha$ emission
line. In a companion paper, we present a similar analysis based on the Fast
Fourier Transform estimate of the power spectrum. We conclude with a comparison
of these two approaches and implications for the upcoming DESI Year 1 analysis.
