The real quadratic Nullstellensatz says:
Let $p(x), q(x)\in \mathbb R[x_1,...,x_n]$ be quadratic polynomials such that $$(*) \ \{x\in \mathbb R^n: p(x)=0\}=\{x\in \mathbb R^n: q(x)=0\}$$ and $p(x)$ has at least one zero $x_0$ such that $grad \ p(x_0) \neq 0$. Then there is a $\alpha \in \mathbb R^*$ such that $p=\alpha q$.
Is the same true if the condition (*) is replaced by the following one: $$ (*) \ \{x\in \mathbb R^n: p(x)=0\} \subset \{x\in \mathbb R^n: q(x)=0\} ? $$