The answer is: It depends. Being quiet or introverted definitely doesn’t preclude you from excelling at management; in fact, you probably bring strengths like introspection and listening skills that can be quite helpful in managing other people. On the other hand, though, if you’re shy to the point of dreading talking to people, that could indeed be a big obstacle to managing well.
And of course, it’s important to point out that being shy and introverted aren’t the same thing. Shy people are generally quiet because they’re nervous or anxious (which can indeed pose difficulties as a manager), while introverts are quiet simply because they’re happier that way, and that’s not inherently a weakness for a manager.
Think about how your personal style is likely to play out in the sorts of tasks a manager’s role is filled with: Are you reasonably comfortable interacting with others? Can you confidently lead a meeting? How do you do with difficult or important professional conversations? As a manager, you’ll have lots of tough conversations, like giving critical feedback or discussing performance concerns (or even firing someone), responding to a request for a raise, dealing assertively with difficult personalities, or delivering bad news (like that a project hasn’t been approved or work needs to be redone). It’s pretty crucial to have these sorts of conversations face to face and not give into temptation to have them over email or otherwise dodge them.
You might also think about whether you might need to put extra effort into ensuring that your employees don’t interpret your shyness as aloofness. For example, you might make a particular point of scheduling regular one-on-ones, taking a clear interest in people’s professional development, and actively soliciting people’s input and opinions. Those three things alone can go a long way toward counteracting any initial impressions of aloofness.
And really, in general, most employees care a lot more about whether their manager is clear, direct, fair, and effective than whether she comes to happy hour. That’s not to say that forming personal bonds doesn’t help – but most people form personal bonds through the act of working closely with people, even if they never talk about their lives outside of work.