People game for different reasons and in different ways. Every group has its own make up that is unique to it. Sure, we have types of gamers and most people fall into the tropes somewhere (that is why they're tropes,) but don't let that fool you into thinking your group is going to act that way all the time.
I say this as a note. This post has advice that is cautionary, but sometimes being quiet is just what the player wants and needs to do to have fun. Sometimes the character is quiet, not the player. That doesn't mean you don't check in, however.
Noticing The Quiet Person
Noticing the quiet person at the table takes a sense of awareness that will develop over time. As a GM you are probably the busiest person at the table. You have 4-6 people telling you stuff, asking you for information, and needing you to resolve actions for them. It is super easy to miss that of the four people at your table only three are talking to you with any sense of regularity. Look for it. Keep track of it. It's worth checking around the table for "other stuff" even when someone just handed you a request.
Approaching The Quiet Person
Once you notice someone is quiet, the next step is just to check in. Don't be overt or on the spot with this. A simple "Is there anything you want to get done here?" that is direct to the player is enough. If they say no, or some other dismissive answer, then you just move on and keep going. If you go more than 20 minutes, or change scenes, and the person is still being quiet, just check in with them again. Include them in your rotations around the table and give them the opportunity to say if they want to do something.
Identifying the Reason
People can be quiet for different reasons (shocking, I know.) Sometimes the player needs to be quiet. They could be sad. They could be down. They could be stressed. They could just not feel like talking. They could be having a blast just watching the antics of the other players. All of these are valid. However, some reasons are what I call "bad" reasons, not because the person shouldn't have them, but because they indicate problems.
What are those reasons? The player feels uncomfortable at the table, the player feels neglected/ignored, the player feels irrelevant, and/or the player isn't having fun. There are more than just those, but you see the pattern. These all mean that the reason the player is being quiet is not only not in character, but directly related to the game or game session. Not good.
After the Game
If you notice someone is quiet most if not all of a session it can be worth talking to them quietly and alone after the game. Make sure they're ok and see if they'll let you know why they're quiet. If not, don't push but let them know they can tell you. Ask them if they're having fun, and what you can do to pull them into the game more or make them have more fun. Maybe they want certain storylines, or are looking for certain hooks? If so, see how they can be incorporated.
The important thing here is you give the player a chance to tell you what is upsetting them (if they're upset) and to also share with you what they want from the game. If they tell you they feel irrelevant or don't know what they can do, see what you - as the GM - can do to help make their character more relevant.
If the player tells you something about what they feel uncomfortable at the table, unsafe, or otherwise awkward around the other players you want to make sure to keep it private (unless they tell you it's ok to share.) You also want to take this seriously. Sometimes serious things are happening at the table. I've had friends feel like they were being harassed or treated unfairly by other players. The internet is full of horror stories of bad players making other players feel unsafe. Take it seriously, listen, look into it, and do what you need to to resolve the situation. No one should feel unsafe at the game table.