In other words, what you are asking is whether every $f\colon\mathbb{H}\to\mathbb{H}$ in $\mathit{SO}_4$ takes the form $x\mapsto \bar u x v$ where $u,v$ are unit quaternions (the connection with your notation is that then $f$ is the composition of the inner automorphism $g\colon x\mapsto \bar v x v$ with left-multiplication by $q = \bar u v$). This is a well-known fact: see, e.g., Conway & Smith, On Quaternions and Octonions (A. K. Peters 2003), §4.1.

Furthermore, this can be seen as an isomorphism $\mathit{Spin}_3\times\mathit{Spin}_3 \to \mathit{Spin}_4$ taking a pair $(u,v)$ of unit octonions (the group of unit octonions is isomorphic to $\mathit{Spin}_3$ acting by conjugation) to $x\mapsto \bar u x v$.

In other words, what you are asking is whether every $f\colon\mathbb{H}\to\mathbb{H}$ in $\mathit{SO}_4$ takes the form $x\mapsto \bar u x v$ where $u,v$ are unit quaternions (the connection with your notation is that then $f$ is the composition of the inner automorphism $g\colon x\mapsto \bar v x v$ with left-multiplication by $q = \bar u v$). This is a well-known fact: see, e.g., Conway & Smith, On Quaternions and Octonions (A. K. Peters 2003), §4.1.

Furthermore, this can be seen as an isomorphism $\mathit{Spin}_3\times\mathit{Spin}_3 \to \mathit{Spin}_4$ taking a pair $(u,v)$ of unit quaternions (the group of unit quaternions is isomorphic to $\mathit{Spin}_3$ acting by conjugation) to $x\mapsto \bar u x v$.