We report the discovery of two iron-group enhanced high-metallicity Mg II absorbers in a search through 28 Ly alpha forest clouds along the PKS 0454 + 039 sight line. Based upon our survey and the measured redshift number densities of W-r(Mg II) less than or equal to 0.3 Angstrom absorbers and Ly alpha absorbers at z similar to 1, we suggest that roughly 5% of Ly alpha absorbers at z less than or equal to 1 will exhibit "weak" Mg II absorption to a 5 sigma W-r(lambda 2796) detection limit of 0.02 Angstrom. The two discovered absorbers, at redshifts z = 0.6248 and z = 0.9315, have W-r(Ly alpha) = 0.33 and 0.15 Angstrom, respectively. Based upon photoionization modeling, the H I column densities are inferred to be in the range 15.8 less than or equal to log N(H I) less than or equal to 16.8 cm(-2). For the z = 0.6428 absorber, if the abundance pattern is solar, then the cloud has [Fe/H] > -1; if its gas-phase abundance follows that of depleted clouds in our Galaxy, then [Fe/H] > 0 is inferred. For the z = 0.9315 absorber, the metallicity is [Fe/H] > 0, whether the abundance pattern is solar or suffers depletion. Imaging and spectroscopic studies of the PKS 0454+039 field reveal no candidate luminous objects at these redshifts. We discuss the possibility that these Mg II absorbers may arise in the class of "giant" low surface brightness galaxies, which have [Fe/H] greater than or equal to -1, and even [Fe/H] greater than or equal to 0, in their extended disks. We tentatively suggest that a substantial fraction of these "weak" Mg II absorbers may select low surface brightness galaxies out to z similar to 1.