Let $p(x) = b_1 x^3 + c_1 x^2 + d_1 x + e_1$ and $q(x) = b_2 x^3 + c_2 x^2 + d_2 x + e_2$. Let us write $a = \frac{r}{s}$ where $r$ and $s$ are relatively prime integers and $s > 0$. Denote $p(a) = m$ and $q(a) = n$. Then
$$b_1 \cdot \frac{r^3}{s^3} + c_1 \cdot \frac{r^2}{s^2} + d_1 \cdot \frac{r}{s} + e_1 = m, \quad b_2 \cdot \frac{r^3}{s^3} + c_2 \cdot \frac{r^2}{s^2} + d_2 \cdot \frac{r}{s} + e_2 = n.$$
Multiplying both identities by $s^3$ we get
$$b_1 r^3 + c_1 r^2 s + d_1 r s^2 + e_1 s^3 = m s^3, \quad b_2 r^3 + c_2 r^2 s + d_2 r s^2 + e_2 s^3 = n s^3.$$
This implies that $s$ divides both $b_1 r^3$ and $b_2 r^3$. Since $s$ and $r$ are relatively prime, we can conclude that $s$ divides $b_1$ and $b_2$. Since $b_1$ and $b_2$ are relatively prime, we have $s = 1$. So, $a = r$ is an integer.