Nicknaming Pokémon rarely has any effect on game-play, and is simply an element of customization that players are free to use or ignore. The only games in which nicknamed Pokémon are treated differently are Pokémon Stadium and Pokémon Stadium 2. In these games, when nicknamed Pokémon transferred from a main series game they have a chance of being alternately colored. This coloration is not the same as being a Shiny Pokémon. This feature has not been included in any later games.
No, nicknames have no effect on the Pokemon's happiness.