Is every metric space quasi-isometric to a graph? I've proved that if $(X, d)$ is a geodesic metric space then there exists a graph which is quasi-isometric to $X$. During this proof I've precisely used the fact that given two point in $X$ there exists a distance minimizing curve inside $X$ joining those two points.
After this I've tried to generalize my proof for arbitrary metric space, which I could not. Now I think it is not true in general. I've an intuitive guess for a counter-example: consider $X=\{x_i \mid i \in \mathbb{N}\}$ and $d(x_i,x_j)= |i^2 - j^2|$. I think for this metric space there dose not exists any graph quasi-isometric with $(X,d)$ but till now I could not able to prove it.
Can anybody provide me some hints or some counter-example for this fact, or may be some proof of the fact that given any metric space there exists a graph quasi-isometric to that space?

A metric space is QI to a connected graph iff it's large-scale geodesic. See Definition 3B1 here (my book with Pierre de la Harpe), and the characterization is an exercise (check the proof of Proposition 3B7(6)).
Large-scale geodesic implies large-scale connected, and your example (the set of squares of integers) is not even large-scale connected. 
