Context. The measurement of the tiny temporal evolution in the redshift of distant objects, the redshift drift, is a powerful probe of universal expansion and cosmology. Aims. We performed the first steps towards the measurement of such an effect using the Lyman-α forest in the spectra of bright quasars as a tracer of cosmological expansion. Our immediate goal is to determine to which precision a velocity shift measurement can be carried out with the signal-to-noise (S/N) level currently available and whether this precision aligns with previous theoretical expectations. A precise assessment of the achievable measurement precision is fundamental for estimating the time required to carry out the whole project. We also aim to study possible systematic effects of an astrophysical or instrumental nature arising in the measurement. Methods. We acquired 12 hours of ESPRESSO observations distributed over 0.875 years of the brightest quasar known, J052915.80-435152.0 (zem = 3.962), to obtain high-resolution spectra of the Lyman-α forest, with a median S/N of ∼86 per 1 km s‑1 pixel at the continuum. We divided the observations into two distinct epochs and analysed them using both a pixel-by-pixel method and a model-based approach. This comparison allows us to estimate the velocity shift between the epochs, as well as the velocity precision that can be achieved at this S/N. The model-based method is calibrated using high-resolution simulations of the intergalactic medium from the Sherwood Simulation Suite, and it provides greater accuracy compared to the pixel-by-pixel approach. Results. We measure a velocity drift of the Lyman-α forest consistent with zero: Δv = ‑1.25‑4.46+ 4.44 m s‑1, equivalent to a cosmological drift of v˙ = ‑1.43‑5.10+5.08 m s‑1 or ż = ‑2.19‑7.78+7.75 × 10‑8 yr‑1. The measurement uncertainties are on par with the expected precision. We estimate that reaching a 99% detection of the cosmic drift requires a monitoring campaign of 5400 hours of integration time over 54 years with an ELT and an ANDES-like high-resolution spectrograph. ⋆ Based on Guaranteed Time Observations collected at the European Southern Observatory under ESO programmes 110.247Q and 112.25K7 by the ESPRESSO Consortium.