Firstly, you can't store a JS object in the database, but you can store JSON.
Secondly, I wouldn't store JSON in that case (and in most cases), because it is terrible to query. I would use a structure along those lines:
users
- id
- name
quizzes
- id
- user_id
questions
- id
- title
answers
- id
- question_id
- title
answer_quiz
- id (this is optional, but with eloquent I always like to have an ID on all tables)
- answer_id
- quiz_id
I don't know what your business rules are (ex. Do you have to be logged in to take the quiz ?), but I think the base structure I laid out above would be a good start, because it is modular and normalized. If you have a new question to add, you don't have to create a migration to add a new column.