The correlation function is the critical ingredient for Quantum Chromodynamics (QCD) sum-rule methods that are used to predict hadronic properties. Thus, in order to perform a sum-rule analysis of hybrids, we need to compute a correlation function involving an operator that probes hybrid states composed to quark-antiquark pair with a gluonic excitation. Using particular combinations of quark and gluon fields and Dirac matrices, we construct currents that probe hybrid states with various J^{PC} quantum numbers. We compute the correlation function to order g_s^3 in QCD, obtaining both perturbative and condensate contributions. The focus here is on light quarkonium and strangeonium hybrids, which involve quark masses small compared to the external momentum scale (m_q^2 << Q^2). While for light quarkonium the calculations are performed in the massless limit, for strangeonium we include a strange quark mass correction to the perturbative result. While the details of the calculations outlined throughout this thesis are outlined for J^{PC} = 0^{+-} and 1^{--} due to interest in the exotic quantum numbers 0^{+-}, ultimately the correlation function is computed for all J^{PC} values with J=0,1. Comparison with existing results for a subset of these J^{PC} quantum numbers provides a validation of our calculations.