This is basically a Pell equation for polynomials. Simply let $\omega_n,\psi_n\in\mathbb{Z}[x]$ such that \[(x+\sqrt{x^2-1})^n=\omega_n+\psi_n\sqrt{x^2-1}.\]Now observe that \[\omega_n^2-\psi_n^2(x^2-1)=(\omega_n+\psi_n\sqrt{x^2-1})(\omega_n-\psi_n\sqrt{x^2-1})=(x+\sqrt{x^2-1})^n(x-\sqrt{x^2-1})^n=\big(x^2-(x^2-1)\big)^n=1\]therefore, $\omega_n$ is a polynomial which satisfies the condition in the statement. Moreover, observe that $\text{deg}(\omega_n)=n$ so for any $n$ there, indeed, exist a polynomial of degree $n$ that satisfies the condition in the statement.