Quantity A
A negative exponent is equal to the reciprocal of the positive exponent.
Thus, $2^{-2} = \dfrac{1}{2^2} = \frac{1}{4}$

$\sqrt{\frac{9}{4}} = \sqrt{\frac{3^2}{2^2}} = \frac{\sqrt{3^2}}{\sqrt{2^2}} = \frac{3}{2}$

Combine to find that $2^{-2} * \sqrt{\frac{9}{4}} = \frac{1}{4} * \frac{3}{2} = \frac{3}{8}$

Quantity B
Again, since a negative exponent is equal to the reciprocal of the positive exponent, $4^{-1} = \dfrac{1}{4^1}$

Sub back in to find that $\frac{2}{3} * 4^{-1} = \frac{2}{3}* \dfrac{1}{4^1} = \frac{2}{12}$


At this point, I'd save time by plugging both into the calculator to find their decimal equivalents to compare the quantities.
Quantity A = 0.375
Quantity B = 0.167

Since 0.375 > 0.167, the answer is A.

If you wanted to compare by making the quantities' denominators equivalent,
Quantity A: $\frac{3}{8} = \frac{3}{8} * \frac{3}{3} = \frac{9}{24}$
Quantity B: $\frac{2}{12} = \frac{2}{12} * \frac{2}{2} = \frac{4}{24}$

Again, since 9 > 4 and their denominators are the same, Quantity A is greater.