This paper reports results obtained from isothermal-isobaric molecular dynamics simulations of binary fluid mixtures composed of one-center Lennard-Jones atoms and two-center Lennard-Jones with point quadrupole molecules. These simulation results are important to investigate the effects of molecular elongation and quadrupole moment on excess properties and to test statistical theories of highly nonideal molecular fluid mixtures, Molecular size and energy parameters of the components are kept to be the same. We show how thermodynamic properties respond to changes in both molecular elongation and quadrupole moment. Simulations are performed over a range of molecular elongation 0 less than or equal to l(D) = L/sigma(D) less than or equal to 0.6 and quadrupole moment 0 less than or equal to q(D) = Q/(epsilon(A) sigma(A)(5))(1/2) less than or equal to 2.5, Results are reported for mixture density, internal energy, excess Gibbs free energy, excess enthalpy and excess volume of 17 equimolar mixtures at a temperature, k(B)T/epsilon(A) = 0.92, and pressure, P sigma(A)(3)/epsilon(A) = 0.5. Excess Gibbs free energy has been determined accurately using the highly reliable coupling parameter charging method, It has been found that excess properties increase as the quadrupole moment increases for a fixed molecular elongation, When the both molecular elongation and quadrupole moment increase simultaneously, there is a strong competition between their effects on the excess properties.