When a metal wire moves across a magnetic field, voltage is induced across it, which is equal to $BLV$, where $B$ is strength of magnetic field, $L$ is the length of wire and $V$ is its velocity. Will a carbon nanotube also exhibit this phenomenon? I am asking this because carbon nanotube is although a good conductor of electricity, but probably that is because of some quantum phenomenon (bulk carbon is bad conductor of electricity).
A carbon nanotube is a conductor.
It is somehow a 'rolled graphene sheet' and the material graphite (stack of graphene layers) is quite a famous conductor.
What characterize the carbon nanotube is its 'fairly' good strength, together with a ballistic conduction.
Therefore as for any conductor, the equation
emf = B.L.V is perfectly valid (the equation reflect an accumulation of charges on both ends of the wire)
Obviously if your wire is too short (let say <0.5um long) then you may have to start considering a bit more your system but with other conductors it would be the same.