While Pokémon games often feature contests, they are typically Beauty, Coolness, Cuteness, and Toughness contests. Bug catching contests were first introduced in Pokémon Diamond and Pearl.
To participate in those contests in later games, you would need to head to the contest area, usually a specific building or location. They are usually timed events and involve catching bugs, with the winner often receiving special prizes.
However, Pokémon Crystal does not include any bug catching contests.