Can you get the exact date a user started following another using the twitter API?

No Ignacio, you can't. You just can know who follows who but not the date the follow started.

Looking at the API, there's is no way, there are two calls to get the followers:

User Methods/statuses/followers

and

Social Graph Methods/followers/ids

Neither of them returns dates or even a serial that would let you see who started following first. Really, there's no indication that twitter is internally storing this information, neither in the API nor Twitter's web interface.

This is a very old question, but perhaps some might be interested to know that while you cannot get the date at which someone started following, you can at least infer an "earliest possible following date" from the fact that the list of followers is ordered according to date, and the fact that follower objects come with a created_at timestamp.

Here's a Python function for calculating an "earliest possible following date": https://github.com/BernhardClemm/twitter-follow-dates

Of course Twitter stores it, because Twitter sorts followers and following lists by the date ;)