Everything is in Lam's book. Theorem III 5.1 says:

All central simple algebra $A$ of dimension $4$ is quaternion. That is $A \cong \left(\frac{a,b}{k}\right)$.

Note that as quadratic space, $A = \langle 1, -a, -b, ab\rangle$ and $V_A = \langle -a, -b, ab\rangle$.

Then Theorem III 2.5 says:

Two quaternion algebras $A, A'$ are isomorphic as $k$-algebra iff $V_A, V_{A'}$ are isomorphic as quadratic spaces.

On the other hand, all quadratic forms of rank three with $d=1$ ($=4^3$ in $k^*/(k^*)^2)$) are of the form $\langle c, d, cd\rangle$. Thus the one-to-one correspondence.

Everything is in Lam's book Introduction to Quadratic Forms over Fields. Theorem III 5.1 says:

All central simple algebra $A$ of dimension $4$ is quaternion. That is $A \cong \left(\frac{a,b}{k}\right)$.

Note that as quadratic space, $A = \langle 1, -a, -b, ab\rangle$ and $V_A = \langle -a, -b, ab\rangle$.

Then Theorem III 2.5 says:

Two quaternion algebras $A, A'$ are isomorphic as $k$-algebra iff $V_A, V_{A'}$ are isomorphic as quadratic spaces.

On the other hand, all quadratic forms of rank three with $d=1$ ($=4^3$ in $k^*/(k^*)^2)$) are of the form $\langle c, d, cd\rangle$. Thus the one-to-one correspondence.