The answer is 1009. First observe that Angela can assign the polynomials $(x-1)(x-2)$, $(x-1)(x-2)(x-3)(x-4)$, $\cdots$, $(x-1)(x-2)\cdots(x-2017)(x-2018)$ (that is, those of even degree) to the group corresponding to $p(x)$, and the others to the group corresponding to $q(x)$. With this choice, the ratio $p(x)/q(x)$ equals
$$\frac{(x-1)(x-2) \cdot (x-1)(x-2)(x-3)(x-4) \cdots}{(x-1) \cdot (x-1)(x-2)(x-3) \cdots} = (x-2)(x-4) \cdots (x-2018),$$
which has degree 1009. To show that a smaller degree cannot be achieved, observe that there are exactly 2017 factors $(x-2)$, exactly 2015 factors $(x-4)$, and in general there is an odd number of factors $(x-k)$ for every even $k$. In order for the polynomial $q(x)$ to divide the polynomial $p(x)$, the number of factors $x-k$ that appear in $p(x)$ must be greater than or equal to the number of factors $x-k$ that appear in $q(x)$: if the total number of factors $x-k$ is odd, in $p(x)$ there must appear at least one more than in $q(x)$. This reasoning shows that in the ratio $p(x)/q(x)$ there must appear at least one factor $(x-2)$, at least one factor $(x-4)$, ..., at least one factor $(x-2018)$, and therefore this ratio cannot have degree less than 1009.