We investigate the effect of the quadrupole-type intramolecular charge transfer (ICT) in open-shell singlet donor-pi-donor (D-pi-D) molecules on the singlet open-shell (diradical) character and the longitudinal second hyperpolarizabilities gamma (the third-order nonlinear optical (NLO) properties at the molecular scale). For this investigation we used the para-quinodimethane (PQM) with point charges (pc's) model calculated with the unrestricted coupled cluster method including single and double excitations with a perturbative treatment of the triple excitations (UCCSD(T)). In this model, the diradical character gamma and the amount of the ICT, that is, the D-pi-D nature, can be varied primarily by changing the exocyclic carbon carbon bond (C C) lengths and the external pc's Q respectively. It turns out that the increase in the D-pi-D nature decreases the gamma values, moves the gamma values (gamma(max)) giving the maximum gamma (gamma(max)) to the large gamma region, and enhances the gamma(max) values, for example, the gamma(max) of the singlet diradical PQM with Q = -2.8 au reaches twice that of the singlet diradical PQM without any pc's. This result indicates that open-shell singlet D-pi-D systems with ICT are promising candidates for a new class of third-order NLO molecules, whose gamma values are more enhanced than those of conventional closed-shell D-pi-D systems and of symmetric open-shell singlet systems without the ICT. To confirm this tendency, we examine the boron-disubstituted PQM dianion model, which is found to exhibit further enhancement of gamma as compared to the PQM model with intermediate diradical character due to the synergy effects of the intermediate open-shell singlet nature and the strong field-induced ICT nature in the dianionic state of the D g D system. Further investigation of the acceptor-pi-acceptor (A-pi-A) type ICT effect in the PQM-pc model shows that both D-pi-D and A-pi-A type symmetric ICTs give similar effects on the relationship between gamma and gamma, though there are some differences originating in the orbital contraction and extension induced by the pc's. The present results contribute to understanding the third-order NLO properties of open-shell symmetric ICT systems and thus to constructing new design guidelines for highly efficient third-order NLO systems.