The “if $V = 0$ then $I$ is maximum” statement does not show a distinctive character of a capacitor and depends on external conditions, namely an exactly sinusoidal waveform without any DC bias.
Indeed, variables should be swapped. When a smooth voltage function $V$ is at its extremum and, hence, has zero time derivative (neither grows nor diminishes), then so does the charge $q$ and, consequently, the current at such moment doesn’t flow. This feature is not restricted to sine waves; continuous differentiability ($C^1$) of $V$ is the requirement, and also the capacitor is assumed to be ideal (it must have no leaks, i.e. the space between and around conductors must have zero conductivity, whereas conductors must have zero voltage drop).
Qualitatively this “current leading by a quarter of period”, not necessarily for a sine waveform, can be explicated with the following table:
$V$ $I$ power
increasing same sign consumed
(by absolute value) as voltage (like a resistor)
extremum 0 0
decreasing opposite sign released
(by absolute value) than voltage (like a battery)