Not much is known about Zelgadiss' family. We know that he is most likely an only child but we know nothing of his parents as he never mentions them in the series. It can be assumed that he doesn't have any, since he appears to have no real home but since he's so determined to find a cure to his current state of being, one could also assume that he doesn't plan on going back until he's returned to his human form.

The one family member that we do know about is Aka Houshi Rezo Greywords, Zelgadiss' great-grandfather. One can assume that the two had a relatively close, trusting relationship with one another at one time, as Zelgadiss most likely would not have gone to Rezo for help otherwise. It would explain why Zelgadiss has developed a very hard time trusting other people afterwards, as a betrayal of a close family member would have been very hard to cope with. It would also explain why Zelgadiss called out to Rezo during their final battle after his transformation into Ruby Eye Shabranigdu and why he felt it necessary to ask Copy Rezo what he would like written on his tombstone; he probably still felt an affinity for Rezo after coming to the conclusion that his actions were not entirely his own.

Although he would probably deny it, I believe that Zelgadiss probably still cared about his great-grandfather once he realised that it was most likely the piece of Shabranigdu sealed inside of him that drove Rezo mad.

I think it's also safe to say that Zelgadiss was never close with many other members of his family, as he himself was unsure if Rezo was his grandfather or great-grandfather. There are also rumors floating around that Rezo murdered Zelgadiss' parents, but there is no concrete evidence of this in any form.