I don't have a counter-example at the moment but the result seems too strong to be true. Here are some partial results instead.
Let $f\in \mathbb{Z}_p[x]$ be so that it's reduction $\pmod{p}$ doesn't have repeated roots. Two matrices $A,B\in GL_n(\mathbb{Z}_p)$, satisfying $f(A)=f(B)=0$, will be conjugate in $\mathbb{Z}_p$ provided that they are conjugate over $\mathbb{F}_p$. This is theorem 2 in Certain matrix equations over rings of integers by R.W. Davis.
If the orders of two matrices $A,B\in GL_n(\mathbb Z/p^n\mathbb Z)$ are coprime to $p$, then they are similar in $GL_n(\mathbb Z/p^n\mathbb Z)$ if and only if their reductions are similar in $GL_n(\mathbb F_p)$. This is proved in the article "On the conjugacy of matrices over a ring of residues" by D.A. Suprunenko (MR0172886).