Let $\displaystyle x\in \mathbb{Q}_p$. We can represent $\displaystyle x$ by the p-adic series $\displaystyle x=\sum_{n\ge k}a_np^n, k,a_n \in \mathbb{Z}, 0\le a_n < p$. Define $\displaystyle f_p:\mathbb{Q}_p \to \mathbb{R}$ by $\displaystyle f_p(x) = \sum_{n \le -k}a_{-n} p^{-n-1}$ (essentially flipping the digits across the decimal point).

Define the relation $\displaystyle \sim$ by $\displaystyle x,y \in \mathbb{Q}_p, x\sim y$ if $\displaystyle f_p(x) = f_p(y)$. Is the quotient topology for $\displaystyle \mathbb{Q}_p / \sim$ metrizable?