Use the y-intercept to find q.

The function is given by $f(x) = p^x + q$.
Point B(0;-3) is the y-intercept of $f$. This means when $x=0$, $f(x)=-3$.
Substitute these values into the equation:

$$-3 = p^0 + q$$

Since any non-zero number raised to the power of 0 is 1 ($p \neq 0$ for an exponential function):

$$-3 = 1 + q$$

Subtract 1 from both sides:

$$q = -3 - 1$$ $$q = -4$$

Point A(3;4) is a point on the graph of $f$. This means when $x=3$, $f(x)=4$.
Substitute $x=3$, $f(x)=4$, and $q=-4$ into the equation $f(x) = p^x + q$:

$$4 = p^3 + (-4)$$ $$4 = p^3 - 4$$

Add 4 to both sides:

$$4 + 4 = p^3$$ $$8 = p^3$$

Take the cube root of both sides:

$$p = \sqrt[3]{8}$$ $$p = 2$$

The values are $p = \boxed{2}$ and $q = \boxed{-4}$.