Let $G$ be a group with a $\mathbb{Q}_p$-representation $\mathfrak{X}: G \to \mbox{GL}_n(\mathbb{Q}_p)$ affording the character $\chi$. We have the $p$-adic norm $\| \cdot \|$ defined on an extension field $K/\mathbb{Q}_p$ generated by the $|G|$-th roots of unity.
I wonder whether $\chi(g)\chi(g^{-1}) = |\chi(g)|^2$ for all $g \in G$ in this situation as is true for $\mathbb{C}$-representations.
It is trivially correct when $n = 1$: the left side is $1$ and the right side is $|\chi(g)|_p^2$, which is $1$ since $\chi(g)$ is a root of unity when $n = 1$.
Is is never true for all $g$ when $n > 1$: at $g = 1$, your desired equation turns into $d^2 = |n|_p^2$, which is incorrect since $n^2$ is an integer greater than $1$ while $|n|_p \leq 1$.
In $\mathbf C$ we can write $|z|^2$ as $z\overline{z}$, but there is nothing like that for $p$-adic fields since $p$-adic absolute values are (positive) real numbers while the values of the characters are in $p$-adic fields. Even in cases where $\chi(g) = 0$, so both sides of your proposed equation are $0$, it would be strange to regard the $p$-adic number $0$ as the real number $|0|_p$. (Yes, we do identify $\mathbf Q$ with its images in $\mathbf R$ and every $\mathbf Q_p$, but I think that is missing my point.)