We present ALMA observations of the GQ Lup system , a young Sun-like star with a substellar mass companion in a wide-separation orbit . These observations of 870 \mu m continuum and CO J=3–2 line emission with beam size \sim 0 \farcs 3 ( \sim 45 AU ) resolve the disk of dust and gas surrounding the primary star , GQ Lup A , and provide deep limits on any circumplanetary disk surrounding the companion , GQ Lup b . The circumprimary dust disk is compact with a FWHM of 59 \pm 12 AU , while the gas has a larger extent with a characteristic radius of 46.5 \pm 1.8 AU . By forward-modeling the velocity field of the circumprimary disk based on the CO emission , we constrain the mass of GQ Lup A to be M _ { * } = ( 1.03 \pm 0.05 ) * ( d / 156 \text { pc } ) M _ { \odot } , where d is a known distance , and determine that we view the disk at an inclination angle of 60 \fdg 5 \pm 0 \fdg 5 and a position angle of 346 \degr \pm 1 \degr . The 3 \sigma upper limit on the 870 \mu m flux density of any circumplanetary disk associated with GQ Lup b of < 0.15 mJy implies an upper limit on the dust disk mass of < 0.04 M _ { \oplus } for standard assumptions about optically thin emission . We discuss proposed mechanisms for the formation of wide-separation substellar companions given the non-detection of circumplanetary disks around GQ Lup b and other similar systems .