Abigail Adams could not have witnessed the first manned hot air balloon flight on November 21, 1783 since she did not arrive in Paris until 1784.
Thomas Jefferson is seen with John Adam and his wife witnessing the first manned hot air balloon flight on November 21, 1783, in Paris, France, but Jefferson was in America at the time and did not arrive in Paris for the first time until 1784. In fact his entire presence there with John Adams prior to 1784 is fictitious.
In Episode 3, a camera is reflected in the carriage window glass during a close-up of young John Quincy Adams just before his departure for Russia. Even the lettering (reversed in the reflection but readable) of the camera's make and model are clearly visible: ARRIFLEX 35.