Recent efforts to characterise the molecular gas content of brightest cluster galaxies (BCGs) at intermediate redshift have revealed a subpopulation of gas-rich systems, whose star formation activity is likely influenced by environmental processing. In this study, we aim to investigate the molecular gas reservoirs and star formation fuelling of central galaxies in groups, also known as brightest group galaxies (BGGs), at intermediate redshifts. We present targeted carbon monoxide (CO) line observations of three BGGs in the COSMOS field at z ∼ 0.3, obtained with the IRAM 30m telescope. The galaxies exhibit disturbed morphologies, extended blue substructures, and interaction signatures. Furthermore, they exhibit significant star formation rates derived from multi-wavelength diagnostics. We detected CO(1̊ightarrow0) emission in one system, revealing a substantial molecular gas mass of M_ H_2 ∼ 3 M_⊙, while for the other two BGGs, CO emission lines remained undetected, yielding stringent upper limits of M_ 10^ 10 H_2 łesssim 10^ 10 M_⊙. By combining molecular gas constraints with fiducial star formation rates derived from total infrared emission, we were able to infer gas depletion timescales in the range łesssim0.5-1.5 Gyr. These results could indicate that despite their active star formation and interaction signatures, some BGGs might already have experienced efficient gas exhaustion or suppressed gas replenishment, suggesting that gas depletion precedes star formation quenching. Our findings hint that environmental processes in galaxy groups could strongly regulate the availability of cold gas and drive rapid evolutionary phases in central galaxies, possibly bridging the gap between gas-rich BCGs and passively evolving systems.