(of a person or animal) extremely quiet or docile.
a shy or excluded person at a dance or party, especially a girl without a partner.
Shrinking violet/Modest violet
A person referred to as a shrinking violet is a timid or shy person.
Reserved
slow to reveal emotion or opinions.
Uncommunicative
unwilling to talk or impart information.
Source: Oxford Dictionaries
In a more general sense, you could use 'introvert', though this is not a specific term. Anyone who keeps things to himself and doesn't disclose his thoughts to many people is an 'introvert'.