We report on a Hubble Space Telescope search for rest-frame ultraviolet emission from the host galaxies of five far-infrared-luminous $zsimeq{}6$ quasars and the $z=5.85$ hot-dust free quasar SDSS J0005-0006. We perform 2D surface brightness modeling for each quasar using a Markov-Chain Monte-Carlo estimator, to simultaneously fit and subtract the quasar point source in order to constrain the underlying host galaxy emission. We measure upper limits for the quasar host galaxies of $m_J>22.7$ mag and $m_H>22.4$ mag, corresponding to stellar masses of $M_ast<2times10^{11}M_odot$. These stellar mass limits are consistent with the local $M_{textrm{BH}}$-$M_ast$ relation. Our flux limits are consistent with those predicted for the UV stellar populations of $zsimeq6$ host galaxies, but likely in the presence of significant dust ($langle A_{mathrm{UV}}ranglesimeq 2.6$ mag). We also detect a total of up to 9 potential $zsimeq6$ quasar companion galaxies surrounding five of the six quasars, separated from the quasars by 1.4-3.2, or 8.4-19.4 kpc, which may be interacting with the quasar hosts. These nearby companion galaxies have UV absolute magnitudes of -22.1 to -19.9 mag, and UV spectral slopes $beta$ of -2.0 to -0.2, consistent with luminous star-forming galaxies at $zsimeq6$. These results suggest that the quasars are in dense environments typical of luminous $zsimeq6$ galaxies. However, we cannot rule out the possibility that some of these companions are foreground interlopers. Infrared observations with the James Webb Space Telescope will be needed to detect the $zsimeq6$ quasar host galaxies and better constrain their stellar mass and dust content.
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