Neuhaeuser et al. (2005) presented direct imaging evidence for a sub-stellar companion to the young T Tauri star GQ Lup. Common proper motion was highly significant, but no orbital motion was detected. Faint luminosity, low gravity, and a late-M/early-L spectral type indicated that the companion is either a planet or a brown dwarf. We have monitored GQ Lup and its companion in order to detect orbital and parallactic motion and variability in its brightness. We also search for closer and fainter companions. We have taken six more images with the VLT Adaptive Optics instrument NACO from May 2005 to Feb 2007, always with the same calibration binary from Hipparcos for both astrometric and photometric calibration. By adding up all the images taken so far, we search for additional companions. The position of GQ Lup A and its companion compared to a nearby non-moving background object varies as expected for parallactic motion by about one pixel (2 pi with parallax pi). We could not find evidence for variability of the GQ Lup companion in the K-band (standard deviation being pm 0.08 mag), which may be due to large error bars. No additional companions are found with deep imaging. There is now exceedingly high significance for common proper motion of GQ Lup A and its companion. In addition, we see for the first time an indication for orbital motion (about 2 to 3 mas/yr decrease in separation, but no significant change in the position angle), consistent with a near edge-on or highly eccentric orbit. We measured the parallax for GQ Lup A to be pi = 6.4 pm 1.9 mas (i.e. 156 pm 50 pc) and for the GQ Lup companion to be 7.2 pm 2.1 mas (i.e. 139 pm 45 pc), both consistent with being in the Lupus I cloud and bound to each other.