First answer so hopefully helpful but I am in a similar situation.
I am fairly new to role-playing but have been background reading for years so have a fair bit of knowledge. I recently joined a group and after a few weeks brought my very quiet 15 year old nephew along — as much to help him socialise (with an s, no matter how much the spell check disagrees) as anything.
The first session he was very quiet so I had to work out ways of engaging him better in the second session.
Common reasons why people are quiet
Shy – they may be afraid to speak, not necessarily an RPG specific issue but a group dynamic issue
Lack of knowledge – they may not know what their options are
Quiet – they may just be quiet and quite happy for other people to take the lead
Slow – they may be trying to formulate a plan but just can't do it very quickly
The first lesson comes from my own recent experience above; I was all about trying to get him more engaged and it turned out he had the time of his life despite me thinking he was ignored all night. So as with all things effective communication is the key — there is no point trying to fix a problem if it is not an actual problem.
Assuming it is a problem I would try to encourage participation, maybe working with the DM to plan things where the others can shine, and if the other players don't speak up make vague suggestions such as "Can you do anything to get us out of this room?", follow them up with more specific questions if you need "Does anyone have x skill that can help us here".
In the first place try not to suggest the full course of action, simply hint at it. This gives people the feeling that when they do act it was their solution not yours.
If this still doesn't work then try and come up with more specific options such as "Can anyone either blow that door up or pick the lock?" and hope someone steps up.
Last step would be to provide actual direction "Locky McLockface, can you please open that door?" which takes away a bit more agency than I like, but people who have difficulty making plans might appreciate it.
Overall it is about making people comfortable that when they come up with a solution it is well received. Never be negative and, at the discretion of the GM, even possibly let bad ideas work just so they don't get that feeling that speaking is useless. Again that is about effective communication with the GM, it is their call but as a player you still have the ability to engage them about it.
Aside from very few instances it boils down to confidence, you can help instill that confidence by teaching, showing, directing and when they do suggest something; trusting. Nothing will put someone back in their shell faster than someone saying "No that is a bad idea".