Quasi-Newton algorithms have the form
$$\begin{align*} \b{d}^{(k)} &= - \b{H}_k \b{g}^{(k)} \br \alpha_k &= \argmin{ \alpha \geq 0 } f (\b{x}^{(k)} + \alpha \b{d}^{(k)}) \br \b{x}^{(k+1)} &= \b{x}^{(k)} + \alpha_k \b{d}^{(k)} \end{align*}$$
where the matrices $\b{H}_1, \b{H}_2, …$ are symetric. In the quadratic case, the above matrices are required to satisfy
$$\b{H}_{k+1} \Delta\b{g}^{(i)} = \Delta\b{x}^{(i)}, 0 \leq i \leq k$$
where $\delta \b{x}^{(i)} = \b{x}^{(i + 1)} - \b{x}^{(i)} = \alpha_i \b{d}^{(i)} $.
Consider a quasi-Newton algorithm applied to a quadratic function with Hessian $\b{Q} = \transpose{ \b{Q}} $, such that for $0 \leq k < n - 1 $,
$$\b{H}_{k+1} \Delta \b{g}^{(i)} = \Delta \b{x}^{(i)}, 0 \leq i \leq k$$
where $\b{H} _{k+1} = \transpose{ \b{H}}_{k+1} $. If $\alpha_i \neq 0, 0 \leq i \leq k+1, $ then $\b{d}^{(0)}, …, \b{d}^{(k+1)} $ are $\b{Q}$-conjugate.
In other words, quasi-Newton methods are also conjugate direction methods.
That also means a quasi-Newton algorithm solves a quadratic of $n $ variables in at most $n $ steps.