Local solubility of a family of ternary conics over a biprojective base I

Let $f,g\in \mathbb{Z}[u_1,u_2]$ be binary quadratic forms. We provide upper bounds for the number of rational points $(u,v)\in {\mathbb{P}}^1(\mathbb{Q})\times {\mathbb{P}}^1(\mathbb{Q})$ such that the ternary conic $$X_{(u,v)}:f(u_1,u_2)x^2+g(v_1,v_2)y^2=z^2$$ has a rational point. We also give some conditions under which lower bounds exist.