These all happen in the Saturday Supercade TV show. To read about other interactions in that show see Saturday Supercade links.
October 1, 1983
Two animated shows based on the Donkey Kong series, called Donkey Kong and Donkey Kong Junior, and one based on the Q*bert series, called Q*bert were produced and aired together in the Saturday Supercade. Additionally, since Q*bert aired right between Donkey Kong and Donkey Kong Junior, the show also featured transition bumpers between them, showing characters from both series meeting each other. These are specifically:
Donkey Kong to Q*bert transition
This clip shows Donkey Kong throwing barrels at Mario and him jumping over them with a pile of barrels behind him. Q*bert then shows up jumping down the pile of barrels, ultimately jumping in his arms and then shaking his hand.
Since both shows were also included in Season 2 in the same order, this is the only clip that was reused.
Q*bert to Donkey Kong Junior transition
This clip shows Q*bert jumping down a staircase and on his flying disk. Donkey Kong Jr. is then shown swinging on the vines, but as Q*bert flies near him, he falls down. Q*bert than flies back and gets Donkey Kong Junior on his disk.
Ruby-Spears was licensed by the games' respective companies (Nintendo and Gottlieb) to produce their cartoons. Once they got the license for both series there was nothing preventing them to make characters from both series appear together.