Yes, it is possible, certainly. But after shyness is removed, the person may remain an introvert. Here, difference between shyness and introversion has to be understood as these two traits are often confused for each other.
A shy person avoids socializing for the fear of going wrong or doing something wrong. Thus, he/she lacks self-confidence what he/she does or intends to do. An introvert has no such feeling of guilt or lack of self-confidence in him/herself, but he/she merely thinks more and speaks less. Thus, an introvert remains happy by him/herself.