This page explains in detail the interactions between the Godzilla series and the Mario series.
The Super Mario Bros. Super Show! episode Mario Meets Koop-zilla
19891019 October 19, 1989
The twenty-eighth episode of the Mario animated series (thirty-fourth also counting the Zelda episodes), is a parody of Godzilla movies, since it's about Mario, Luigi, Princess Toadstool and Toad visiting the oriental city of Sayonara to find Super Sushi, capable of making anyone who eats it grow gigantic, but once there they find out that King Koopa has already eaten it. After some failed attempts at stopping him, Mario eats Super Sushi too, growing gigantic, fighting back Koopa and allowing Luigi and Toad to inject him with Super Sushi Shrinker, making him return to his regular size and run away.
King Koopa's gigantic transformation is an obvious reference to the monster Godzilla, and the way he fights Mario is a reference to giant monsters clashes. Also, the city of Sayonara is based on a generic japanese city, referencing most Godzilla movies being set in Japan.
The Adventures of Super Mario Bros. 3 episode Dadzilla
19901006 October 06, 1990
The ninth episode of The Adventures of Super Mario Bros. 3 is about Kootie Pie and Big Mouth, two of King Koopa's children, going to Hollywood searching for a new dad, and they find one by turning to life a giant robotic monster used to shoot scenes for a movie, later revealed by Luigi to be named "Madzilla". Once Madzilla starts causing trouble in Hollywood, Mario, Luigi, Princess Toadstool and Toad try to stop him, and after some failed attempts they finally build a robot version of Madzilla's mom using parts from various giant robot monsters, including the original head of Godzilla's mom. By talking to Madzilla and to the two Koopalings as their grandmother, Mario controlling the robot they built manages to make the two kids return to the Mushroom Kingdom, and finally offer Madzilla to stay there too and start a new life.
The title of the episode and the creature Madzilla are obviously based on Godzilla, but additionally his mother, according to Luigi featured in the movie Revenge of Madzilla's Mom, may be a reference to the movie Son of Godzilla, featuring Godzilla involved in a father-son relationship.
Mario in Mariozilla
February - March 1996
This two-part comic published in the german Club Nintendo magazine is about Mario accidentaly growing gigantic and traveling to New York in search for help by Dr. Light from the Mega Man series (see Mario X Mega Man#Mario in Mariozilla for more details).
The title of the comic is clearly a reference to Godzilla, however apart from involving a giant living being, the plot has nothing to do with Godzilla, and instead features a scene reminiscent of King Kong (see King Kong X Mario#Mario in Mariozilla).
DiC, that created the Mario TV shows, and Nintendo of Germany, that published Mario in Mariozilla, hold no right over Toho's Godzilla, in fact they only parodied it.