"If the question is rectricted to positive definite forms, then a C exists. ".
If $C$ is allowed to depend on the form, it is quite simple. Shifting to get rid of linear terms, completing the square and multiplying by an appropriate integer, one can reduce the problem to showing that for every positive integers $b$ and $D$ the quadratic form $x^2+by^2$ cannot represent too many successive integers divisible by $D$. Now just choose a prime $p>D$ such that $-b$ is not a quadratic residue modulo $p$ (you do not need Dirichlet theorem or even quadratic reciprocity law to find such prime though these two tools make its existence obvious) and notice that the form cannot represent any number divisible by $p$ but not by $p^2$. But such numbers are fairly dense among the ones divisible by $D$.
The interesting question is whether one can find a universal $C$ that would work for every quadratic polynomial of 2 variables with integer coefficients and positive definite quadratic part.