= Solution
Every degree-three monomial evaluated at $(1,f,g)$ is $f^ag^b$ with $a+b\leq3$. Its possible pole orders at $p,q,r$ are at most $a$, $a+b$, $b$, respectively, and hence at most three each. Therefore $Q(1,f,g)\in L(3p+3q+3r)$.
The <vector space> of homogeneous degree-three polynomials in three variables has <dimension> $\binom52=10$. The target <Riemann-Roch space> has <dimension> nine, because its <divisor on an algebraic curve> has degree nine and <genus> is one. Thus the evaluation <linear map>
$$
Q\longmapsto Q(1,f,g)
$$
has nontrivial kernel. Consequently
$$
\boxed{\text{there is a nonzero homogeneous cubic }Q\text{ with }Q(1,f,g)\equiv0.}
$$
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