If I play from hidden a Teemo Strategist during a Showdown and one of the cards that I look at is a Nocturne, can I play it directly at the battlefield in question?

Yes, you can play Nocturne directly to the battlefield in question, provided you control that battlefield.

According to the established rulings for these cards:

• Playing Nocturne: Nocturne's ability allows you to play him when he is looked at or revealed from the top of your deck. Card text supersedes normal timing restrictions. While units normally cannot be played during a "showdown closed" state, Nocturne's specific triggered ability allows him to be played into any battlefield his controller controls.
• Control Requirement: You can play Nocturne to the battlefield you are defending as long as you control it. In almost all cases where you are defending with Teemo, you retain control of that battlefield for the duration of the showdown.
• Interaction: If you play Teemo, Strategist from Hidden during a showdown and his ability reveals Nocturne, you may choose to play Nocturne using his ability instead of recycling him.

Nuance: If you are in a rare situation where you are defending but do not control the battlefield (such as a "surprise defense" where you moved into an open showdown), you would not be able to play Nocturne to that battlefield.