When you link up with another GAMEBOY via link cable in Pokemon and enter the Link Club, you get sent to a room with a table in the middle. Once you are done with the trading or battling, is there any other way to leave this room other than turning off and on the GAMEBOY console?
No, there's not.
From Gen 2 onward this process was made more user-friendly, but in the original first-gen games the only way to escape the Cable Club is to reset.
There is actually a way to escape the Cable Club without resetting, called the Cable Club Escape Glitch. The most common method of performing it is by bringing only a poisoned Pokemon into the Cable Club and walking around until it faints and your character blacks out, which will kick you back into the main Pokemon Center map. However, due to the various effects of this glitch:
After exploiting the glitch, the player is unable to save under normal circumstances because the save button is replaced by the reset button. The player is also unable to open the item pack, because normally items cannot be used when in link with another player. Provided that these settings are kept, if the player proceeds to encounter any Trainer outside of the Cable Club their roster will be replaced with the one last stored by the game when trading or battling with the player, and their picture will be replaced with Red's.
It is not viable as an actual, useful method of bypassing the need to reset.