I'm interning at a software company in which I'm viewed as "quiet." However, I just choose not to partake in the usual frivolous office gossip. Additionally, it's hard to follow the conversations when there are many inside jokes, etc. I'm doing great work and I know that they really like my results, but I do not want them to think of me as a quiet person, because in reality I am quite outgoing. What can I do?
Honestly, don't put so much weight in what they think. You're just interning so they will ALWAYS say something about you in order to make themselves feel like they deserve whatever the main positions or whatever else more than you - it just sounds like common demonization so that they can keep you lower on the totem pole.
My advice, do your internship, stay awesome in the way you are, then find somewhere that you fit in naturally and don't worry what others 'say' about you, especially in an internship position. If they are keeping you around it means they don't actually feel poorly about you so it doesn't matter.
The simple solution: If you don't want to be seen as quiet, talk more.
However, I would like to point out that your work personality and your outside-work personality do not have to match, and, in fact, often will differ significantly. For example, you may drink more socially than you would in a work setting, or tell off-color jokes among friends that you wouldn't say at work. By choosing not to partake in what you view as "frivolous office gossip", you're cultivating a work image that is more serious and more quiet than your coworkers -- and they've noticed. This seems to be a deliberate choice on your part, as you seem not to view the gossip as an activity you wish to partake in. It's unlikely to work against you; better to be seen as too serious than too frivolous, and better to be seen as too quiet than too loud. So are you sure you want to change it?
If you still do, you're going to have to engage with your coworkers more. Where you're currently seeing pointless chatter, they're probably seeing valuable social experience: the chance to bond and get to know each other outside of the routine of work. Try joining in some of these conversations -- it will give you a chance to get to know them and them to get a feel for what you're like. Furthermore, you'll start to pick up on the inside jokes more the more you chat with them.