We present the results of infrared L-band (3-4\ensuremathμm) slit spectroscopy of 30 PG QSOs at z < 0.17, a representative sample of local high-luminosity, optically selected AGNs. The 3.3\ensuremathμm polycyclic aromatic hydrocarbon (PAH) emission feature is used to probe nuclear (< a few kpc) starburst activity and to investigate the connections between AGNs and nuclear starbursts in PG QSOs. The 3.3\ensuremathμm PAH emission is detected in the individual spectra of 5/30 of the observed PG QSOs. We constructed a composite spectrum of PAH-undetected PG QSOs and discerned the presence of the 3.3\ensuremathμm PAH emission therein. We estimated the nuclear-starburst and AGN luminosities from the observed 3.3\ensuremathμm PAH emission and 3.35\ensuremathμm continuum luminosities, respectively, and found that the nuclear-starburst-to-AGN luminosity ratios in PG QSOs are similar to those of previously studied AGN populations with lower luminosities, suggesting that AGN-nuclear starburst connections are valid over a wide luminosity range of AGNs in the local universe. The observed nuclear-starburst-to-AGN luminosity ratios in PG QSOs with available supermassive black-hole masses are comparable to a theoretical prediction based on the assumption that the growth of a supermassive black hole is controlled by starburst-induced turbulence.