Let's say that the point around which to rotate ${\color{Orange} o}$ has coordinates $\left( {\color{Orange} o_{{\color{Red} x}}},\, {\color{Orange} o_{{\color{Green} y}}},\, {\color{Orange} o_{{\color{Blue} z}}} \right)$, that the point to rotate ${\color{Purple} v}$ has the coordinates $\left( {\color{Purple} v_{{\color{Red} x}}},\, {\color{Purple} v_{{\color{Green} y}}},\, {\color{Purple} v_{{\color{Blue} z}}} \right)$ and thus the Quaternion Representation ${\color{Purple} v} ~\widehat{=} \begin{pmatrix} {\color{Purple} v_{{\color{Black} w}}}\\ {\color{Purple} v_{{\color{Red} x}}}\\ {\color{Purple} v_{{\color{Green} y}}}\\ {\color{Purple} v_{{\color{Blue} z}}} \end{pmatrix} \widehat{=}~ {\color{Purple} v_{{\color{Black} w}}} + {\color{Purple} v_{{\color{Red} x}}} \cdot \mathrm{i} + {\color{Purple} v_{{\color{Green} y}}} \cdot \mathrm{j} + {\color{Purple} v_{{\color{Blue} z}}} \cdot \mathrm{k}$ has, that the point is rotated to ${\color{Purple} v_{\text{rotaded}}}$ has coordinates $\left( {\color{Purple} v_{\text{rotaded},\, {\color{Red} x}}},\, {\color{Purple} v_{\text{rotaded},\, {\color{Green} y}}},\, {\color{Purple} v_{\text{rotaded},\, {\color{Blue} z}}} \right)$ and thus has the quaternion representation ${\color{Purple} v} ~\widehat{=} \begin{pmatrix} {\color{Purple} v_{\text{rotaded},\, {\color{Black} w}}}\\ {\color{Purple} v_{\text{rotaded},\, {\color{Red} x}}}\\ {\color{Purple} v_{\text{rotaded},\, {\color{Green} y}}}\\ {\color{Purple} v_{\text{rotaded},\, {\color{Blue} z}}} \end{pmatrix} \widehat{=}~ {\color{Purple} v_{\text{rotaded},\, {\color{Black} w}}} + {\color{Purple} v_{\text{rotaded},\, {\color{Red} x}}} \cdot \mathrm{i} + {\color{Purple} v_{\text{rotaded},\, {\color{Green} y}}} \cdot \mathrm{j} + {\color{Purple} v_{\text{rotaded},\, {\color{Blue} z}}} \cdot \mathrm{k}$ and that the target point ${\color{Cyan} a}$ has the coordinates $\left( {\color{Cyan} a_{{\color{Red} x}}},\, {\color{Cyan} a_{{\color{Green} y}}},\, {\color{Cyan} a_{{\color{Blue} z}}} \right)$ and thus has the quaternion representation ${\color{Cyan} a} ~\widehat{=} \begin{pmatrix} {\color{Cyan} a_{{\color{Black} w}}}\\ {\color{Cyan} a_{{\color{Red} x}}}\\ {\color{Cyan} a_{{\color{Green} y}}}\\ {\color{Cyan} a_{{\color{Blue} z}}} \end{pmatrix} \widehat{=}~ {\color{Cyan} a_{{\color{Black} w}}} + {\color{Cyan} a_{{\color{Red} x}}} \cdot \mathrm{i} + {\color{Cyan} a_{{\color{Green} y}}} \cdot \mathrm{j} + {\color{Cyan} a_{{\color{Blue} z}}} \cdot \mathrm{k}$.
Since the only thing that differentiates ${\color{Purple} v_{\text{rotaded}}}$ and ${\color{Cyan} a}$ is the absolute distance to ${\color{Orange} o}$, which we can normalize:
$$
\begin{align*}
{\color{Purple} v_{\text{rotaded}}} &= ||{\color{Purple} v_{\text{rotaded}}}|| \cdot {\color{Purple} v_{\text{rotaded},\, 0}}\\
{\color{Cyan} a} &= ||{\color{Cyan} a}|| \cdot {\color{Cyan} a_{0}}\\
{\color{Purple} v_{\text{rotaded},\, 0}} &= {\color{Cyan} a_{0}}\\
||{\color{Purple} v}|| &= ||{\color{Purple} v_{\text{rotaded}}}||\\
{\color{Purple} v_{\text{rotaded}}} &= ||{\color{Purple} v}|| \cdot {\color{Cyan} a_{0}}\\
{\color{Purple} v_{\text{rotaded}}} &= ||{\color{Purple} v}|| \cdot \frac{{\color{Cyan} a}}{||{\color{Cyan} a}||}\\
\end{align*}
$$
Thus, the target point ${\color{Purple} v_{\text{rotaded}}}$ in the form of the point to be drawn ${\color{Cyan} a}$ and the origin point can be represented ${\color{Purple} v}$.
The easiest way I could think of to do this would be to bring the orange dot to the center of the coordinate system and then use the classic formula for $3$D rotation of a vector via quaternions.
Then we can take ${\color{Orange} o}$ as the coordinate origin by subtracting the components of ${\color{Orange} o}$ from the points and thus moving everything relative to to the center:
$$
\begin{align*}
\mathrm{v} &= \begin{pmatrix} 0\\ {\color{Purple} v_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ {\color{Purple} v_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ {\color{Purple} v_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} \widehat{=} \left( {\color{Purple} v_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}} \right) \cdot \mathrm{i} + \left( {\color{Purple} v_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}} \right) \cdot \mathrm{j} + \left( {\color{Purple} v_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \right) \cdot \mathrm{k}\\
\mathrm{v}_{\text{rotaded}} &= \begin{pmatrix} 0\\ {\color{Purple} v_{\text{rotaded},\, {\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ {\color{Purple} v_{\text{rotaded},\, {\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ {\color{Purple} v_{\text{rotaded},\, {\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} \widehat{=} \left( {\color{Purple} v_{\text{rotaded},\, {\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}} \right) \cdot \mathrm{i} + \left( {\color{Purple} v_{\text{rotaded},\, {\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}} \right) \cdot \mathrm{j} + \left( {\color{Purple} v_{\text{rotaded},\, {\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \right) \cdot \mathrm{k}\\
\mathrm{v}_{\text{rotaded}} &= \begin{pmatrix} 0\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} \widehat{=} \left( \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}} \right) \cdot \mathrm{i} + \left( \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}} \right) \cdot \mathrm{j} + \left( \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \right) \cdot \mathrm{k}\\
\mathrm{a} &~= \begin{pmatrix} 0\\ {\color{Cyan} a_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ {\color{Cyan} a_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ {\color{Cyan} a_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} \widehat{=}~ \left( {\color{Cyan} a_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}} \right) \cdot \mathrm{i} + \left( {\color{Cyan} a_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}} \right) \cdot \mathrm{j} + \left( {\color{Cyan} a_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \right) \cdot \mathrm{k}\\
\end{align*}
$$
We can apply this to the standard formula for quaternion rotation $\mathrm{v}_{\text{rotated}} = \mathrm{q} \cdot \mathrm{v} \cdot \overline{\mathrm{q}}$ and get:
$$
\begin{align*}
\begin{pmatrix} 0\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} &\widehat{=} \begin{pmatrix} q_{w}\\ q_{{\color{Red} x}}\\ q_{{\color{Green} y}}\\ q_{{\color{Blue} z}} \end{pmatrix} \cdot \begin{pmatrix} 0\\ {\color{Purple} v_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ {\color{Purple} v_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ {\color{Purple} v_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} \cdot \begin{pmatrix} q_{w}\\ -q_{{\color{Red} x}}\\ -q_{{\color{Green} y}}\\ -q_{{\color{Blue} z}} \end{pmatrix}\\
\begin{pmatrix} 0\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ \frac{||{\color{Purple} v}||}{||{\color{Cyan} a}||} \cdot {\color{Cyan} a_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} &\widehat{=} \begin{pmatrix} \cos\left( \frac{\varphi}{2} \right)\\ q_{{\color{Red} x}}\\ q_{{\color{Green} y}}\\ q_{{\color{Blue} z}} \end{pmatrix} \cdot \begin{pmatrix} 0\\ {\color{Purple} v_{{\color{Red} x}}} - {\color{Orange} o_{{\color{Red} x}}}\\ {\color{Purple} v_{{\color{Green} y}}} - {\color{Orange} o_{{\color{Green} y}}}\\ {\color{Purple} v_{{\color{Blue} z}}} - {\color{Orange} o_{{\color{Blue} z}}} \end{pmatrix} \cdot \begin{pmatrix} \cos\left( \frac{\varphi}{2} \right)\\ -q_{{\color{Red} x}}\\ -q_{{\color{Green} y}}\\ -q_{{\color{Blue} z}} \end{pmatrix}\\
\end{align*}
$$
So what you are looking for is probably the quaternion $q = q_{w} + q_{{\color{Red} x}} \cdot \mathrm{i} + q_{{\color{Green} y}} \cdot \mathrm{j} + q_{{\color{Blue} z}} \cdot \mathrm{k}$ where $\varphi$ is the angle between ${\color{Cyan} a}$ and ${\color{Purple} v}$ to ${\color{Orange} o}$.
