The following corollary is from Discrete groups by Ohshika.
Corollary 2.70. Hyperbolic groups are finitely presented.
The author didn't prove it but said that 'Combining this theorem with the preceding proposition, we get the following'. Here's the theorem and proposition:
Proposition. Let $G$ be a $\delta$-hyperbolic group, and suppose that $d\geq 4\delta+2$. Then the Rips complex $P_d(G)$ is contractible and locally finite.
Theorem. Let $G$ be a hyperbolic group. Then $G$ acts on $P_d(G)$ properly discontinuosuly, and simplicially. Moreover, $P_d(G)$ has finite dimension, and the quotient $P_d(G)/G$ is compact.
Rips complex here is defined as follows:
Let $X$ be a metric space, and let $d$ be a non-negative real number. The Rips complex $P_d(X)$ is a simplicial complex such that the set of vertices is equal to $X$, and $p+1$ points $x_0,x_1,\ldots,x_p$ span a $p$-simplex if and only if the diameter of $\{x_0,\ldots,x_p\}$ is less than or equal to $d$.
Now suppose there is a hyperbolic group $G$ which is not finitely presented. Then the presentation of $G$ has infinitely many relations. In particular, the length of the relations is unbounded. Let $d = 4\delta+2$ and choose a relation $r$ whose length is strictly bigger than $d$. Then we can find a simplicial sphere contained in $P_d(G)$ so that $P_d(G)$ is not contractible which contradicts the proposition.
Does my argument make sense? I mean I only used Proposition here.
Your proof by contradiction cannot possibly work because for any group and any nonempty generating set, there are infinitely many relations. For example, for any generator $g$ and any $n \ge 1$ the word $g^n g^{-n}$ is a relator. For something more substantial, if $R_1,R_2$ are relators and $w_1,w_2$ are any words, then $w_1^m R_1 w_1^{-m} w_2^n R_2 w_2^{-n}$ are relators for any $m,n \ge 1$. As for where your proof breaks down, it's last sentence Then we can find a simplicial sphere... does not make much sense to me.
I wouldn't bother using a proof by contradiction.
Here's a direct proof for a special case, namely when $G$ is torsion free. From that assumption, it follows that the action of $G$ on $P_d(G)$ is free. You can then apply covering space theory together with simple connectivity of $P_d(G)$ to conclude $P_d(G)/G$ is a finite complex whose fundamental group is isomorphic to $G$. Now use standard methods of algebraic topology read off a finite presentation of $G$ from $P_d(G)/G$.