John Adams looks at the famous painting of the signing of the Declaration of Independence and says that everyone in the painting is dead except for Thomas Jefferson and himself. But Charles Carroll, who is pictured, was a signer from Maryland who died in 1832, living 6 years longer than Adams.
'Nabby' Adams was diagnosed with breast cancer in 1811 not 1803 as shown in the series and underwent a mastectomy in 1812. She died in 1814 from the rapidly spreading cancer.
The name of Thomas Jefferson's mistress and slave, Sally Hemings, is misspelled in the closing credits of this episode, employing two m's instead of one. The closing credits list the character as Sally Hemmings, when in fact her surname was Hemings.
John Quincy Adams refers to his "State of the Union Address." The event did not receive that name until at least 1934, previously having been called "The President's Annual Message to Congress"; furthermore, for many years, including this one, it was delivered not as an address but a written communication.
When Thomas Jefferson writes his response to John Adams' letter, he uses a writing frame to hold his pen, held up by a modern rubber band. That kind of rubber band wouldn't be invented for at almost two decades after the death of both Adams and Jefferson.