Step 1: Let the radius of the circle be $r$. Area of the circle $=\pi r^{2}$.
Step 2: A quadrant is a sector of angle $90^{\circ}$, i.e. one-fourth of the circle.
$$\text{Area of quadrant}=\frac{90^{\circ}}{360^{\circ}}\times\pi r^{2}=\frac{1}{4}\pi r^{2}$$
Step 3: Required ratio
$$=\frac{\frac{1}{4}\pi r^{2}}{\pi r^{2}}=\frac{1}{4}=1:4$$
Correct option is (C) $1:4$.