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In fact it is not true in this generality: the simple counterexample, inspired by a famous Aesop's fable, is: $X=\mathbb{R}$, $f(x)=x$, and $g(x)=\max(0, 2x-1)$ for $x\in[0,1]$. The quotient distance is clearly zero: for $0<\epsilon< 1$ the homeomorphism $ \phi_\epsilon :[0,1]\ni x \mapsto \max(\epsilon x, 2x-1)\in[0,1]$ gives $\|f\circ \phi_\epsilon-g\|_\infty<\epsilon$. But of course for any couple of homeomorphisms $\phi$ and $\psi$ from $[0,1]$ to itself we have $f\circ\phi\neq g\circ\psi$, as the former is injective and the latter is not.