This is actually a terrible idea for a lot of reasons concerning tech, licensing, bandwidth, and the sheer and utter camp of watching a streaming video through the game client. Additionally, now that neftlix has the rights to CBS content and will be offering Star Trek episodes streaming in April I'd recommend that you organize 'Party View' with your fleet and watch those episodes together that way.
Think about the absurdity of your suggestion: Log into Star Trek Online to watch TV.
Are Star Trek episodes really coming to Netflix in April? I'd be majorly happy if they were.
as WoW has showed us, Social Zones are used for Trolling and Exchanging/Selling. I like the way STO has handled this as most people in Social Zones, as well as the forums, are generally kind, friendly, and helpful.