Gibbs simulation techniques have been used to calculate vapor-liquid equilibria of monatomic molecules having both dipoles and linear quadrupoles embedded in their Lennard-Jones cores. Results presented here include the coexistence curves and the estimates of the critical temperatures and densities for a range of values of both Q*2 and mu*2, where mu*2=mu2/(epsilonsigma3)=1.00 or 2.00 and Q*2=Q21(epsilonsigma5) = 1.00, 1.50, 2.00, or 2.50. Our simulation results indicate that both T(c)* and rho(c)* increase with increasing values of the electrostatic moments, and both are more sensitive to Q*2 than mu*2.