Does the Legend of Zelda version available from the Wii's Virtual Console service feature the "second Quest" that was accessible once you either completed the game once (or put ZELDA as your name when you started the game)?
According to Wikipedia:
The Virtual Console release of The Legend of Zelda (NES) uses the updated version featured in 2003's The Legend of Zelda: Collector's Edition compilation for the GameCube. While the gameplay is identical to the 1986 original, this release includes the save screen from the Famicom Disk System version, as well as an updated translation of the introduction screen.
Furthermore, I can find no reference stating that the "Second Quest" has been removed. I would logically conclude that if there is no reference regarding it being taken away, it should be in the game.
We retro-gamers never take modifications to original games lightly.