Is it possible to establish trade routes in "one city challenge" since you only have one city?
You never "build" trade routes; you build harbors and roads. In OCC, you still can build harbors and roads. If you mean, "can I form a trade route between two cities?", the answer is no, because you only have one city and cannot form trade routes with other Civs in Civ5 (unlike civ4). You can still connect city states to you by road for the quest, but calling that a trade route is inaccurate, as it gives only a one-time bonus.
AI civs will still gain the effects of trade routes.
The simple answer is "NO", as you can only build trade routes between your own cities. However, if you are willing to look at mods there is one,
Which lets you trade between allies.