We have obtained optical intermediate-resolution spectra (R=3000) of the narrow-line quasars DMS 0059-0055 and PG 1543+489. The [O III] emission line in DMS 0059-0055 is blueshifted by 880 km s^-1 relative to Hbeta. We also confirm that the [O III] emission line in PG 1543+489 has a relative blueshift of 1150 km s^-1. These two narrow-line quasars show the largest [O III] blueshifts known to date among type 1 active galactic nuclei (AGNs). The [O III] emission lines in both objects are broad (1000-2000 km s^-1), and those in DMS 0059-0055 show strong blue asymmetry. We interpret the large blueshift and the profile of the [O III] lines as the result of an outflow interacting with circumnuclear gas. Among type 1 AGNs with large blueshifted [O III], there is no correlation between the Eddington ratios and the amount of [O III] blueshift. Combining our new data with published results, we confirm that the Eddington ratios of such AGNs are the highest among AGNs with the same black hole mass. These facts suggest that the Eddington ratio is a necessary condition for [O III] blueshifts or that the [O III] blueshifts weakly depend on the Eddington ratio. Our new sample suggests that there are possibly other necessary conditions to produce an outflow than a high Eddington ratio: a large black hole mass (>10⁷ M_solar), a high mass accretion rate (>2 M_solar yr^-1), or a large luminosity [lambdaL_lambda(5100Å)>10^44.6 ergs s^-1].