"Is X part of group Y?" tends to be a difficult question to answer when there are multiple definitions for group Y; West Virginia is frequently classified as Southern despite splitting off from Virginia to join the Union. There is no strict definition of the Founding Fathers; there were a lot of men centrally involved in the transition from thirteen interconnected colonies under the British Crown to thirteen states banded together under a single government. However, the general idea is men who signed the Declaration of Independence, the Articles of Confederation, and/or the Constitution.
It's important to remember also that the men who participated in this process were not friends. Oh, some of them were, but they were different people, with different occupations (although most of them had legal knowledge, and by the end practically all of them had served in government one way or another), and different interests. They quarreled over the power of the federal government, both in terms of its general ability to require things of the states and the people, and actual specific things that it would be allowed to do. They fought over commerce in the colonies, and to what degree which American industries needed to be protected from established manufacturing of goods overseas -- British lumber and French wine were a threat to American lumber and wine. And, of course, they fought over slavery, though many northern men still owned slaves, working in their houses. So when people invoke "what the Founding Fathers meant/wanted", there isn't really any such thing; they were not a monolithic group that agreed on everything, or even many things.
Anyway, we've got something of a definition of the Founding Fathers; did John Quincy Adams sign any of those documents? Of course not. Quincy was only nine years old when the Declaration was written, and twenty-one when the Constitution was ratified. He was with his father John Adams learning how American government was going to work for much of this period, but that doesn't make him influential in the process.