We present ALMA [C II] 158 $mu$m line and far-infrared (FIR) continuum emission observations toward HSC J120505.09$-$000027.9 (J1205$-$0000) at $z = 6.72$ with the beam size of $sim 0.8 times 0.5$ (or 4.1 kpc $times$ 2.6 kpc), the most distant red quasar known to date. Red quasars are modestly reddened by dust, and are thought to be in rapid transition from an obscured starburst to an unobscured normal quasar, driven by powerful active galactic nucleus (AGN) feedback which blows out a cocoon of interstellar medium (ISM). The FIR continuum of J1205$-$0000 is bright, with an estimated luminosity of $L_{rm FIR} sim 3 times 10^{12}~L_odot$. The [C II] line emission is extended on scales of $r sim 5$ kpc, greater than the FIR continuum. The line profiles at the extended regions are complex and broad (FWHM $sim 630-780$ km s$^{-1}$). Although it is not practical to identify the nature of this extended structure, possible explanations include (i) companion/merging galaxies and (ii) massive AGN-driven outflows. For the case of (i), the companions are modestly star-forming ($sim 10~M_odot$ yr$^{-1}$), but are not detected by our Subaru optical observations ($y_{rm AB,5sigma} = 24.4$ mag). For the case of (ii), our lower-limit to the cold neutral outflow rate is $sim 100~M_odot$ yr$^{-1}$. The outflow kinetic energy and momentum are both much smaller than what predicted in energy-conserving wind models, suggesting that the AGN feedback in this quasar is not capable of completely suppressing its star formation.
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