Is Hilbert's tenth problem decidable for degree $2$?

I quote from page $1$ of the same notes:

A Diophantine problem over $mathbb{Q}$ is concerned with the solutions either in $mathbb{Q}$ or in $mathbb{Z}$ of a finite system of polynomial equations $$F_i(X_1, ldots , X_n) = 0 hspace{0.5in} (1 leq i leq m) hspace{1.0in} (1)$$ with coefficients in $mathbb{Q}$. Without loss of generality we can obviously require the coefficients to be in $mathbb{Z}$. A system $(1)$ is also called a system of Diophantine equations. Often one will be interested in a family of such problems rather than a single one; in this case one requires the coefficients of the $F_i$ to lie in some $mathbb{Q}(c_1, ldots , c_r)$, and one obtains an individual problem by giving the $c_j$ values in $mathbb{Q}$. Again one can get rid of denominators. Some of the most obvious questions to ask about such a family are:

(A) Is there an algorithm which will determine, for each assigned set of values of the $c_j$, whether the corresponding Diophantine problem has solutions, either in $mathbb{Z}$ or in $mathbb{Q}$?

(B) For values of the $c_j$ for which the system is soluble, is there an algorithm for exhibiting a solution?

For individual members of such a family, it is also natural to ask:

(C) Can we describe the set of all solutions, or even its structure?

I quote from pages $13$ to $14$ of these progress notes on Diophantine equations:

The most important invariant of a curve is its genus. In the language of algebraic geometry over $mathbb{C}$, curves of genus $0$ are called rational, $ldots$ A canonical divisor on a curve $Gamma$ of genus $0$ has degree $−2$; hence by the Riemann-Roch Theorem $Gamma$ is birationally equivalent over the ground field to a conic. The Hasse principle holds for conics, and therefore for all curves of genus $0$; this gives a complete answer to Question (A) at the beginning of these notes. But it does not give an answer to Question (B). Over $mathbb{Q}$, a very simple answer to Question (B) is as follows:

Theorem 1 Let $a_0, a_1, a_2$ be nonzero elements of $mathbb{Z}$. If the equation $$a_0 {X_0}^2 + a_1 {X_1}^2 + a_2 {X_2}^2 = 0$$ is soluble in $mathbb{Z}$, then it has a solution for which each $a_i {X_i}^2$ is absolutely bounded by $|a_0 a_1 a_2|$.

Siegel has given an answer to Question (B) over arbitrary algebraic number fields, and Raghavan has generalized Siegel’s work to quadratic forms in more variables.

The knowledge of one rational point on $Gamma$ enables us to transform $Gamma$ birationally into a line; so there is a parametric solution which gives explicitly all the points on $Gamma$ defined over the ground field. This answers Question (C).