No, John Adams didn’t have much influence, if any at all, in his son’s Presidency.  The elder Adams was 89 years old when John Quincy Adams was inaugurated and in poor health at the time; he was unable to even attend the inauguration and died exactly 16 months after JQA’s inauguration.  You have to remember that communication didn’t travel as quickly, either, so it’s not like John Adams could pick up the phone or even send a telegram to John Quincy.

I think JQA visited Massachusetts during his Presidency before his father died, but John Quincy was his own man, so I don’t think he could have been influenced by his father at that point, and I doubt John Adams would have tried.  By that time, John Quincy had earned his wings, and John Adams had been so far removed from the Presidency and Washington politics that he would have been totally out-of-the-loop.  And, like you said, he was old and tired and probably had no desire to jump back into the arena even if he could have.