Quark doesn't have any children of his own (goodness knows he'd be paying out the nose for child support if he did) but he helped raise Nog. In a lot of ways, Nog is very much his kid.
When Quark was a kid, his father wasn't a good businessman. We learn over the course of the show that he's a lot more like his mother. She's brilliant with managing finances and gaining profit. Rom is much more like their father and for that reason Quark initially keeps him on a short leash. Rom's already been screwed over by his first wife and Quark feels like he has to protect his brother, even as he also tries to be a good ferengi by exploiting him.
Nog, raised with his father and uncle as his primary guardians, is acutely aware that his father has no real business sense. And while Nog is his father's son in many ways... he's also Quark's son. I doubt Quark would appreciate thinking it's his influence on Nog that leads him to join Starfleet, but Nog does inherit his uncle's intelligence and realizes that while he doesn't want the same life his father has... he doesn't want the life Quark has, either. Chasing profit and having it never be truly fulfilling enough. Nog's smart enough to figure out what he wants his own path to be, however, and he applies the lessons of their culture in ways that might not gain Quark's approval but does often gain his respect. (And inspires his father to eventually chart his own path too.)
Quark may never outright say it, but the way he feels at heart is pretty clearly that Nog is his kid too; we see him act parental, concerned... perhaps treating Nog at times the same way Ishka treated Quark and Rom when they were younger. Quark takes an interest in Nog's schooling when he's young and while he thinks Starfleet Academy is a mistake, still takes pride in being part of Nog's send off. Quark often 'mothers' Nog; I'm not really sure how else to put the way Quark will hover over Nog when he worries about his nephew, but it's definitely behavior Quark learned from his own mother. And when Nog is injured so badly during the war that he loses his leg, Quark nurses his nephew and even sets aside his own ideals to pick up a weapon in defense of them both from the attacking jem'hadar.
Every time I watch this show I think I come to love the way Quark, Rom, and Nog grow and change as a family over the course of the years a little more.
