Theoretically, you could use Twitter's developer API for this. There is a query that will return a given user's followers, which you can then inspect (possibly using the query to convert an ID to a full user object to fill out extra information if needed) to determine if your game's user ID is among the set.
Once you find your game's user ID among the player's followed users, you know they've followed the account you want and can reward them accordingly.
Naturally, you will need to get the user to give you their Twitter ID in order for this to work. Since the results of the API are likely eventually-consistent, you may need to implement the confirmation as a periodic poll while your game is running, rather than expecting it to return the correct data immediately after the user comes back from the "follow us on Twitter" flow.
Note that I have no idea if this sort of usage falls into what is or is not acceptable to Twitter. The front page of their API says one use-case is "[evaluating] Twitter data to inform business decisions," which this could conceivably be, but the examples under that category don't really match.
You should read over Twitter's terms of use for their developer API, and have your lawyer do the same with an understanding of your specific business case. I'd check the restrictions section, particularly the part about "off-Twitter matching."
