We have performed a coupled electron–nuclear dynamics study of H₂⁺ molecular ions under the influence of an intense few-cycle 4.5 fs laser pulse with an intensity of 4 × 10¹⁴ W/cm² and a central wavelength of 750 nm. Both quantum and classical dynamical methods are employed in the exact similar initial conditions with the aim of head-to-head comparison of two methodologies. A competition between ionization and dissociation channel is explained under the framework of quantum and classical dynamics. The origin of the electron localization phenomena is elucidated by observing the molecular and electronic wave packet evolution pattern. By probing with different carrier envelope phase (CEP) values of the ultrashort pulse, the possibility of electron localization on either of the two nuclei is investigated. The effects of initial vibrational states on final dissociation and ionization probabilities for several CEP values are studied in detail. Finally, asymmetries in the dissociation probabilities are calculated and mutually compared for both quantum and classical dynamical methodologies, whereas Franck–Condon averaging over the initial vibrational states is carried out in order to mimic the existing experimental conditions.