relationship between Gaz and Zim in the show could be accurately described as unfriendly and aloof. Acquaintances at best, neither character has ever expressed anything close to affection for or even a friendly demeanour towards the other. In fact, Gaz has hardly spoken more than a few lines to Zim, and clearly considers him a vague nuisance rather than an actual threat to Earth.
Their contact throughout the series is very limited. Not unlike Dib's disregard for GIR, Zim often brushes his mortal enemy's little sister off when dealing with the pair of them. It is hinted in Future Dib that he thinks she's "scary" and might be intimidated by her, which is not unlikely, considering that Gaz frightens most children and adults she comes into contact with. And despite Zim's very real capability to obliterate the human race, Gaz finds him too "dumb" to take over the world, as she stated in Gaz, Taster of Pork.
In NanoZIM, Gaz beat up Zim in what she thought was a online video game. It is notable that she recognized Zim's voice. However, his voice is pretty easy to distinguish, and she's probably used to hearing it considering all the fighting between Zim and Dib.
In The Wettening, Zim held Gaz above his head to protect himself from the rain. Gaz probably would have beaten him up if Dib didn't intervene with his puddle splash.
In Planet Jackers, Gaz doesn't seem to care that Zim is in her house, but rather is annoyed about Dib yelling about it.
In Bloaty's Pizza Hog, Dib asks Gaz if she even cares that Zim is trying to destroy all of mankind. Gaz claims that Zim is bad at it. Later on in the episode she fights Zim in space and seems to wave at him. This, of course, is done in a relatively mocking fashion.
In Tak: The Hideous New Girl, Gaz claims that Zim's voice is "making her sick", and then sprays him with soda. Later when Zim takes her and Dib to his base, he threatens to "melt Gaz's face off or something" if she touches anything - he pauses before saying this, though most likely because he was trying to think up a proper threat.

In Gaz, Taster of Pork, Zim refused to help fix Gaz's ailment, though the fact that Dib was asking for the assistance probably had a lot to do with this. Later in the episode, Gaz tells the Shadowhog how she knows that Zim is an alien but how she also believes he is too dumb to take over the Earth.
In the unmade episode Ten Minutes to Doom, Gaz only grudgingly helps Zim to retrieve his PAK from Dib. Note that, despite the fact that Zim will die without his PAK, she only decides to help him when he tricks her into thinking Dib also stole her Gameslave. Later in the episode, upon discovering that Zim was the one who actually stole it, Gaz chases after him with the likely intent of physically abusing him.
In the unmade episode Mopiness of Doom, Gaz apparently wasn't at all bothered by Zim disappearing from skool for a few weeks; in fact, she was much more concerned with how bizarre Dib was acting.