It seems to me like the proof for Abelian categories works here too. Consider the following diagram where $d = (id,-id)$, the middle vertical map is the fold (or $+$) map, and the square on the right is a homotopy pushout:

$$ \begin{array}{ccccc}
   X & \xrightarrow{d} & X \oplus X & \xrightarrow{(f,g)} & Y \\
   \downarrow & & \downarrow & & \downarrow \\
   0 & \to & X & \to & C \\
   \end{array}
$$

From the second square being a pushout, $C$ is the coequalizer of $f$ and $g$. If we check the first square is a pushout too, we get that the outer rectangle is a pushout and so $C$ is also $\mathrm{cofib}(f-g)$.