Edit: I have clarified a bit more the relationship between conic bundles and quaternion algebras and the relationship to weak approximation, and tidied up some typos.

I have myself recently started studying problems of this type, and it turns out that there is much to be said and such questions lead to very rich research problems. As far as I am aware, the first person to study such problems in any kind of generality was Serre [1].

I work in the following setting: Let $B$ be a projective variety over $\mathbb{Q}$ with a Zariski dense set of rational points and let $X$ be a non-singular variety together with a flat morphism $\pi:X \to B$ such that every fibre is a non-singular projective curve of genus zero. Then it is well-known that such a curve is isomorphic to a plane conic, and flatness implies that all the fibres are plane conics. Hence we obtain a conic bundle. To such a conic bundle one may asssociate an element $Q_\pi$ of $Br_2(\mathbb{Q}(B))$. If for example the generic fibre of $\pi$ is given by
$$ax^2 + by^2 = z^2, (*)$$
where $a,b \in \mathbb{Q}(B)$, then $Q_\pi$ is the class given by the quaternion algebra $(a,b)$. This construction is essentially a bijection, namely given a quaternion algebra $(a,b)$ over $\mathbb{Q}(B)$ one may associate to it in an obvious way the conic bundle $(*)$. This element controls most of the arithemtic of the family. One can show that $Q_\pi$ is trivial if and only if $\pi$ admits a rational section (we will use this fact later to address Felipe's question). Following
Serre, I instead consider the limit
$$C(B,\pi,H) = \limsup_{T \to +\infty} \frac{\big\{ b \in B(\mathbb{Q}) \mid H(b) \leq T, \, X_b(\mathbb{Q}) \neq \emptyset \big\}}{\big\{ b \in B(\mathbb{Q}) \mid H(b) \leq T \big\}},$$
where $H$ is a choice of height function on $B$.

Serre [1] basically only considers the case where $B = \mathbb{P}^n$. Let $H$ be the usual naive height on $\mathbb{P}^n$. Then Serre shows that
$$C(\mathbb{P}^n, \pi, H)=0, $$
as soon as there exists a divisor $D$ in $\mathbb{P}^n$ such that the residue of $Q_\pi$ at $D$ is non-trivial (he in fact shows something more precise, see [1]).
In particular in such cases, "most" conics in the family do not contain a rational point. Moreover, one knows that an element $A \in Br(\mathbb{Q}(T_1,..,T_n))$ admits a non-zero residue if and only if $A \not \in Br(\mathbb{P}^n_{\mathbb{Q}}) = Br(\mathbb{Q})$. In particular, we see that $C(\mathbb{P}^n,\pi,H) >0$ if and only if $Q_\pi=0$, which by the above remark happens if and only if $\pi$ obtains a rational section (which in particular implies that we must have $C(\mathbb{P}^n,\pi,H)=1$). I believe this answers your question in this case. Your specific conic bundle
$$qx^2 -y^2 = z^2, (**)$$
corresponds to the class of the quaternion algebra $(t,-1)$ over $Br(\mathbb{Q}(t))$. In this case, Serre's more general result in fact shows that
$$\#\{q \in \mathbb{Q}: H(q) \leq T, (**) \text{ has a solution}\}
\ll \frac{T^2}{\log T}.$$

Here is how one may obtain conic bundles which do not admit rational sections, but for which $C(B,\pi,H) =1$. Suppose that $B$ admits a Brauer group element $A \in Br(B)$ which is the class of a non-trivial quaternion algebra, for which $A \otimes_{\mathbb{Q}_v} \mathbb{Q}_v = 0$ for every place $v$ of $\mathbb{Q}$ (such elements may e.g. be given by elements of the Tate-Shaferavich groups of algebraic tori, see Sansuc's classical paper [2]). As by assumption $A \otimes_{\mathbb{Q}_v} \mathbb{Q}_v = 0$ for all $v$, we see that every conic in the family is everywhere locally soluble. The Hasse-Minkowski theorem therefore implies that every conic in the family contains a rational point, hence $C(B,\pi,H) =1$. However, since $A$ was non-trivial we see that $\pi$ does not admit a rational section.

Your question in the general case is highly non-trivial. Assuming that the results proved by Serre hold in any kind of generality, one would expect that $C(B,\pi,H)=0$ if $Q_\pi \not \in Br(B)$. To obtain positive values of $C(B,\pi,H)$ which are not $1$, one would need a quaternion algebra $A$ over $\mathbb{Q}(B)$ for which there exists some place $v$ of $\mathbb{Q}$ and some $b_1,b_2 \in B(\mathbb{Q})$ such that $A_v(b_1) \neq 0 \in Br(\mathbb{Q}_v)$ but $A_v(b_2) = 0 \in Br(\mathbb{Q}_v)$. This would force some conics in the family to fail to have $\mathbb{Q}_v$-points (hence not have $\mathbb{Q}$-points), but at least make sure that there are also some conics in the family which still have $\mathbb{Q}_v$-points (hence could still have $\mathbb{Q}$-points). Such elements are exactly the ones which give rise to a failure of weak approximation for $B$ via the Brauer-Manin obstruction. So the values of $C(B,\pi,H)$ which may be obtained should intimately depend on the extent to which $B$ fails weak approximation, at least if the Brauer-Manin obstruction is the only one to weak approximation (for more details on the Brauer-Manin obstruction, see [2]).

Problems of this type naturally admit numerous generalistions e.g. to Brauer-Severi varieties and to number fields, where many of the above remarks still hold true.

[1] Serre - Spécialisation des éléments de $Br_2(\mathbb{Q}(T_1,..,T_n))$.

[2] Sansuc - Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.