We report on deep near-infrared F125W (J) and F160W (H) Hubble Space Telescope Wide Field Camera 3 images of the z = 6.42 quasar J1148+5251 to attempt to detect rest-frame near-ultraviolet emission from the host galaxy. These observations included contemporaneous observations of a nearby star of similar near-infrared colors to measure temporal variations in the telescope and instrument point-spread function (PSF). We subtract the quasar point source using both this direct PSF and a model PSF. Using direct subtraction, we measure an upper limit for the quasar host galaxy of m(J) > 22.8 and m(H) > 23.0 AB mag (2 sigma). After subtracting our best model PSF, we measure a limiting surface brightness from 0.'' 3 to 0.'' 5 radius of mu(J) > 23.5 and mu(H) > 23.7 AB mag arcsec(-2) (2 sigma). We test the ability of the model subtraction method to recover the host galaxy flux by simulating host galaxies with varying integrated magnitude, effective radius, and Sersic index, and conducting the same analysis. These models indicate that the surface brightness limit (mu(J) > 23.5 AB mag arcsec(-2)) corresponds to an integrated upper limit of m(J) > 22-23 AB mag, consistent with the direct subtraction method. Combined with existing far-infrared observations, this gives an infrared excess log(IRX) > 1.0 and corresponding ultraviolet spectral slope beta > -1.2 +/- 0.2. These values match those of most local luminous infrared galaxies, but are redder than those of almost all local star-forming galaxies and z similar or equal to 6 Lyman break galaxies.