What is the phase difference between wave P and wave Q?

Wave P starts at its maximum positive displacement at $t=0$. This corresponds to the peak of a cosine function.
Wave Q starts at zero displacement and is increasing at $t=0$. This corresponds to the start of a sine function.

A standard sine wave, $y = A \sin(\omega t)$, has a phase of 0 at $t=0$.
A wave that starts at its maximum positive displacement, like wave P, is equivalent to a sine wave shifted by $\frac{1}{4}$ of a cycle (or $\frac{\pi}{2}$ radians) to the left. So, wave P has an initial phase of $+\frac{\pi}{2}$ relative to wave Q.

The phase difference ($\Delta \phi$) between wave P and wave Q is the difference in their initial phases.

Alternatively, observe that wave P is shifted by one-quarter of a wavelength (or period) relative to wave Q. Since one full cycle corresponds to $2\pi$ radians, a quarter cycle corresponds to:

The calculated phase difference is $\frac{\pi}{2}$, which matches option A.

The final answer is $\text{A}$.