Extended Lyα emission is ubiquitously observed around quasars. This emission traces the cool gas within the circumgalactic medium and provides insights into the complex interplay between gas dynamics and active galactic nucleus feedback. However, its connection to the cold molecular gas of the host galaxies remains largely unexplored. We aim to characterize the molecular gas reservoirs in quasars at cosmic noon and investigate how they are linked to extended Lyα emission. We present ALMA observations of the CO(4-3) transition in 37 quasars at z from the QSO MUSEUM survey, which have been previously mapped in Lyα with VLT/MUSE. We derive molecular gas masses and gas fractions, explore correlations with Lyα nebula and quasar properties, and search for CO-emitting companions in the fields. Of 37 quasars, 21 are detected in CO(4-3), with gas masses of M_ ≈(3-40) . Quasars with the most massive molecular gas reservoirs are associated with the centrally dimmest Lyα nebulae, while those hosting the centrally brightest Lyα nebulae are generally not detected in CO. This suggests that gas and dust in the hosts regulate Lyα escape and consequently affect the emission from halo gas. We find evidence that quasars with lower Eddington ratios harbor more massive gas reservoirs, while strongly accreting quasars (łambda_ ⪆ 0.9) likely deplete their gas; for example, through powerful quasar-driven outflows. Despite their higher molecular gas masses within the sample, the low-Eddington quasars with CO detections exhibit low gas fractions with a median of M_ /M_* ∼ 0.10, below what is typically found for inactive star-forming galaxies. Six quasars are marginally resolved in CO, with effective radii up to ∼ 8, . In addition, we detect 14 companion galaxies at high fidelity, indicating an overall overdense environment in the quasar fields with a derived quasar-galaxy cross-correlation length of 9.81^ gas M_⊙ Edd gas kpc +2.22 _ -2.05 -1 cMpc .