We present first results from Multi Unit Spectroscopic Explorer (MUSE) observations at the Very Large Telescope in the MUSE Ultra Deep Field (MUDF), a $approx 1.2times 1.4$ arcmin$^2$ region for which we are collecting $approx$200 hours of integral field spectroscopy. The $approx 40$-hour observation completed to date reveals the presence of a group of three Ly$alpha$ nebulae associated with a bright quasar pair at $zsimeq3.23$ with projected separation of $approx 500rm~kpc$. Two of the nebulae are physically associated with the quasars which are likely powering the Ly$alpha$ emission, and extend for $gtrsim 100~rm kpc$ at a surface brightness level of $approx 6times 10^{-19}~rm erg~s^{-1}~cm^{-2}~arcsec^{-2}$. A third smaller ($approx$35 kpc) nebula lies at a velocity offset of $approx 1550$ km s$^{-1}$. Despite their clustered nature, the two large nebulae have properties similar to those observed in isolated quasars and exhibit no sharp decline in flux at the current depth, suggesting an even more extended distribution of gas around the quasars. We interpret the shape and the alignment of the two brighter nebulae as suggestive of the presence of an extended structure connecting the two quasar host galaxies, as seen for massive galaxies forming within gas-rich filaments in cosmological simulations.