The quantum dynamics of vibrational excitation and dissociation of H-2(+) by strong and temporally shaped infrared (IR) laser pulses has been studied on the femtosecond (fs) time scale by numerical solution of the time-dependent Schrodinger equation with explicit treatment of nuclear electron motion beyond the Born-Oppenheimer approximation. Using sin(2)-shaped laser pulses of 120 fs duration with a peak intensity of I-0 > 10(14) W/Cm-2, it has been found that below-resonant vibrational excitation with a laser carrier frequency of omega < omega(10)/2 (where omega(10) is the frequency of the vertical bar nu = 0 > -> vertical bar nu = 1 > vibrational transition) is much more efficient than a quasi-resonant vibrational excitation at omega approximate to omega(10). In particular, at the below-resonant laser carrier frequency omega = 0.3641 x 10(-2) au (799.17 cm(-1)), dissociation probabilities of H-2(+) (15.3% at the end of the 120 fs laser-pulse and 21% at t = 240 fs) are more than 3 orders of magnitude higher than those obtained for the quasi-resonant laser frequency omega = 1.013 X 10(-2) au (2223.72 cm(-1)). Probabilities of State-selective population transfer to vibrational states vertical bar nu = 1 >, vertical bar nu = 2 >, and vertical bar nu = 3 > from the vibrational ground state vertical bar nu = 0 > of about 85% have been calculated in the optimal below-resonant cases. The underlying mechanism of the efficient below-resonant vibrational excitation is the electron-field following and simultaneous transfer of energy to the nuclear coordinate.