quotient of finitely presented module $\DeclareMathOperator\Coker{Coker}$Assume the exact sequence
$$A \xrightarrow{f} B \xrightarrow{} \Coker f \xrightarrow{} \{0\} $$ where $A$ is a finitely generated module and $B$ is a finitely presented module.
Is it true that $\Coker f$ is finitely presented module ?
In general, is the quotient of a finitely presented module to a finitely generated submodule, finitely presented ?

This is true.  
I'll denote the ring by $R$, while remaining silent about whether these are left or right modules.
Write $B = R^n/I$ for some finitely generated submodule $I \subseteq R^n$.  Then the image of $f \colon A \to B$ is of the form $J/I$ for some submodule $J \subseteq R^n$, and $J/I$ is a finitely generated module.  Then $J$ is finitely generated as well: taking finitely many elements of $J$ whose images generate $J/I$, along with a finite generating set for $I$, will give a finite generating set for $J$.  
Thus $Coker(f) \cong R^n/J$ is finitely presented.
If you're looking for a reference, see Exercise 4.8 in T.Y. Lam's book Exercises in Modules and Rings.
