We study, the Coulomb scattering lifetimes of electrons in two coupled quantum wires by calculating the quasiparticle self-energy within a multisubband model of quasi-one-dimensional (Q1D) electron system. We consider two strongly coupled quantum wires with two occupied subbands. The intrasubband and intersubband inelastic scattering rates are caculated for electrons in different subbands. Contributions of the intrasubband, intersubband plasmon excitations, as well as the quasiparticle excitations are investigated. Our results show that the plasmon exictations of the first subband are the most important scattering mechanism for electrons in both subbands.