I would like to compute the divisor class group of the projective quadric cone $$ Q=\mathrm{Proj}(\mathbb{C}[X_0,X_1,X_2,X_3]/(X_1X_2-X_3^2)). $$ It has as an open subset the quadric cone $U$ in $\mathbb{C}^3$, which, according to Example II.6.5.2 of Hartshorne's, satisfies $\mathrm{Cl}(U)\simeq \mathbb{Z}/2\mathbb{Z}$. Therefore we have an exact sequence: $$ \mathbb{Z}\rightarrow \mathrm{Cl}(Q)\rightarrow \mathbb{Z}/2\mathbb{Z}\rightarrow 0, $$ where the first map sends $1$ to the class of the divisor $Q\setminus U$, and the second map is restriction.
I am not sure if this helps, but it is all I have been able to do so far.
By the way, what would be the canonical class of $Q$? Could we use the adjunction formula to compute it?
Let $X = Proj(\mathbb{C}[X_1,X_2,X_3]/(X_1 X_2 - X_3^2))$. Now, observe that $Q = \overline{C(X)}$, where $C(X)$ is the affine cone over $X$ and bar above it denotes the projective closure. Now use the exercise 6.3(a) of Hartshorne's Algebraic Geometry, Chapter 2 to get
$$Cl(Q) \cong Cl(X)$$
Now clearly $X \cong \mathbb{P}^1$ via the veronese embedding. Thus $Cl(X) \cong Cl(\mathbb{P}^1) \cong \mathbb{Z}$.
The exact sequence written above has an interpretation in the following form
$$\mathbb{Z} \rightarrow Cl(Q) \rightarrow Cl(C(X))$$
This is well-described in exercise 6.3(b).