I have edited this answer to add in a reference to a result of Bieri which essentially proves that the groups in this answer are not finitely presentable. This result is quite general and may be applied in lots of different settings. If I had more time then I would completely rewrite the answer to focus on this result and make Rips' construction an application of it, but alas I do not have the time so instead am writing this preamble...
Examples of finitely generated, non-finitely presentable groups can be found via Rips' construction (E. Rips, Subgroups of small Cancellation Groups,
Bull. Lond. Math. Soc., 14 (1982), 45–47 (doi)):
Let $Q$ be the group defined by your favourite presentation. Then there exists a hyperbolic* group $G$ and a short exact sequence $1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1$ where $N=\langle a, b\rangle$ is finitely generated. You care about this sequence because if gives you subgroups of hyperbolic groups with pathological properties (e.g. take $Q$ to have insoluble word problem, then $N$ has insoluble membership problem, while as the triviality problem is insoluble for groups (so we do not know if $Q$ is trivial) the generation problem is insoluble for hyperbolic groups (we do not know if $G=\langle a, b\rangle=N$)). However:
Fact. $Q$ is infinite if and only if $N$ is finitely presentable.
So the bad subgroups of hyperbolic groups which we found are all non-finitely presentable.
The proof of the fact is non-trivial, and follows from three facts:
In general, if a group $H$ has cohomological dimension two and $K\lhd H$ is a finitely presented normal subgroup of $H$ then $K$ is free or of finite index in $H$. (This is Corollary 8.6 in Bieri, Robert. "Homological dimension of discrete groups." Queen Mary College Mathematics Notes (1976).)
The group $G$ in Rips' construction has cohomological dimension $2$ (as it is torsion-free small cancellation).
The subgroup $N$ is normal but not free.
*In fact $C'(\lambda)$ for your favourite $\lambda\leq1/6$.