Using the spectral norm, we have the following optimization problem in matrix $\mathrm X \in \mathbb R^{n \times n}$
$$\begin{array}{ll} \text{minimize} & \| \mathrm A \mathrm X - \mathrm X \mathrm B \|_2\\ \text{subject to} & \mathrm X^\top \mathrm X = \mathrm I_n\end{array}$$
where matrices $\mathrm A, \mathrm B \in \mathbb R^{n \times n}$ are given. Introducing variable $t \in \mathbb R$ and rewriting in epigraph form,
$$\begin{array}{ll} \text{minimize} & t\\ \text{subject to} & \| \mathrm A \mathrm X - \mathrm X \mathrm B \|_2 \leq t\\ & \mathrm X^\top \mathrm X = \mathrm I_n\end{array}$$
Note that $\| \mathrm A \mathrm X - \mathrm X \mathrm B \|_2 \leq t$ is equivalent to
$$t^2 \mathrm I_n - \left( \mathrm A \mathrm X - \mathrm X \mathrm B \right)^\top \left( \mathrm A \mathrm X - \mathrm X \mathrm B \right) \succeq \mathrm O_n$$
Dividing both sides by $t > 0$,
$$t \, \mathrm I_n - \left( \mathrm A \mathrm X - \mathrm X \mathrm B \right)^\top \left( t \, \mathrm I_n \right)^{-1} \left( \mathrm A \mathrm X - \mathrm X \mathrm B \right) \succeq \mathrm O_n$$
Using the Schur complement test for positive semidefiniteness, we obtain the following optimization problem in matrix $\mathrm X \in \mathbb R^{n \times n}$ and scalar $t > 0$
$$\begin{array}{ll} \text{minimize} & t\\ \text{subject to} & \begin{bmatrix} t \, \mathrm I_n & \mathrm A \mathrm X - \mathrm X \mathrm B\\ \left( \mathrm A \mathrm X - \mathrm X \mathrm B \right)^\top & t \, \mathrm I_n\end{bmatrix} \succeq \mathrm O_{2n}\\ & \mathrm X^\top \mathrm X = \mathrm I_n\end{array}$$
Relaxing the equality constraint $\mathrm X^\top \mathrm X = \mathrm I_n$, we obtain $\mathrm X^\top \mathrm X \preceq \mathrm I_n$, a linear matrix inequality (LMI) that can be rewritten as follows
$$\begin{bmatrix} \mathrm I_n & \mathrm X\\ \mathrm X^\top & \mathrm I_n\end{bmatrix} \succeq \mathrm O_{2n}$$
Thus, the relaxation of the original optimization problem is a (convex) semidefinite program (SDP) in matrix $\mathrm X \in \mathbb R^{n \times n}$ and scalar $t > 0$
$$\begin{array}{ll} \text{minimize} & t\\ \text{subject to} & \begin{bmatrix} t \, \mathrm I_n & \mathrm A \mathrm X - \mathrm X \mathrm B & & \\ \left( \mathrm A \mathrm X - \mathrm X \mathrm B \right)^\top & t \, \mathrm I_n & & \\ & & \mathrm I_n & \mathrm X\\ & & \mathrm X^\top & \mathrm I_n\end{bmatrix} \succeq \mathrm O_{4n}\end{array}$$
It remains to be determined whether this relaxation is actually useful.