He can run for congress, but not the senate. There are only three requirements for standing for election.
1. He must be a resident of the state he wants to represent.
2. He must be 25 for congress (30 for the senate)
3. He must be a citizen for the past 7 years. I don't know if you can be a naturalized citizen under the age of 21. Article 1 Section 8 says that congress has the right to determine immigration laws, but it says nothing about age. Currently the law is that citizenship is granted to those applying who are over the age of 18. I am therefore assuming that John Quincy is living now and is not a historical figure. He must have filled out form N - 600.
He may also have become a citizen if one of his parents was a US citizen or if they were naturalized while he was a minor. Immigration laws are extremely complex. I'm assuming that he became a US citizen by applying for it using form N - 600 after the age of 18.