I'm aware about the fact stated above, but I'm not able to find some references or proofs besides Gromov's Hyperbolic Groups - Essays in Group Theory. I'll state things precisely. I will consider a complete geodesic metric space $(X,d)$. Given $x,y\in X$, we denote any short geodesic joinning $x$ and $y$ by $[x,y]$.
Definition: a complete geodesic metric space $(X,d)$ is said to be Gromov $\delta$-hyperbolic if every geodesic triangle whose vertices are $x_1,x_2,x_3$ satisfy the following condition: the distance between any $x\in [x_i,x_j]$ and $[x_i,x_j]\cup[x_j,x_k]$ is less than $\delta$, where we take index $i,j,k$ mod.3
Definition: A complete geodesic metric space $(X,d)$ is (A,B)-quasi-convex if any two geodesic segments $\alpha,\beta:[0,1]\rightarrow X$ satisfy the following condition: $$d^H(\alpha([0,1]),\beta([0,1]))\leqslant A\max\{d(\alpha(0),\beta(0)),d(\alpha(1),\beta(1))\}+B$$ where $d^H$ is the Hausdorff distance.
Theorem: If $(X,d)$ is a complete geodesic metric space and is $\delta$-hyperbolic, then $(X,d)$ is $(A,B)$-quasi convex for some $A,B$.
Does anyone know some reference for the proof? It seems to be a well-known fact on the context. Any help will be really appreciated.
The upper bound, though I have not seen it written in precisely this form, is a quick consequence of the thinness of triangles. Here is something from A Course in Metric Geometry by Burago, Burago, Ivanov:
The authors omit the proof, which is easy: $[ab]$ is in the $\delta$-neighborhood of $[ac]\cup [bc]$, and the latter set is in the $d(b,c)$-neighborhood of $[ac]$.
Applying the above lemma twice (moving endpoints of geodesic one at a time), you'll get $$d^H(\alpha([0,1]),\beta([0,1])) \leqslant d(\alpha(0),\beta(0)) + d(\alpha(1),\beta(1)) +2\delta$$ which gives the right hand side with $A=2$, $B=2\delta$.
The left hand side cannot hold as stated: what if $\alpha,\beta$ are the same long geodesic path traveled in different directions?