Go to Settings>Mobile

Text message notifications:
Twitter will send alerts to your phone, depending on your settings. You can limit it to particular times (I would not want to receive alerts in the middle of the night), or particular types of messages. You can limit to only those you allow to send to send alerts via twitter (perhaps, your family, dearest friends, or your boss, if you tweet for work); to direct messages, as well as mentions and replies.

So, you’ll want to think about how many times you want twitter to alert your phone and for what. If you do not want twitter to alert you for any reason, make sure all of the notification messages are UNCHECKED.