The dialog on HBO is obviously dramatised, but Washington did privately profess to disapprove of such titles. In a letter to David Stuart dated 26 July 1789, Washington wrote that:
It is to be lamented that [Adams] and some others have stirred a question, which has given rise to so much animadversion, and which I confess has given me much uneasiness, lest it should be supposed by some, unacquainted with facts, that the object they had in view was no displeasing to me.
The reason Washington gave was that he "foresaw and predicted" the hostile reception John Adams' proposals were met with. Adams himself wrote, albeit years later, that Washington:
had conversed with me before on the Subject And I know that his opinion was in favour of a Title . . . Washington was far from expressing any disapprobation of a Title; So far that he thought a decent Tittle to the Office to be useful and proper but he wanted no Title himself and was convinced from what he heard in Conversation and read in the public Papers, that if the Title of Highness was given it would excite such a popular Clamour, that Congress would either be obliged to rescind it.
Whether Washington would still disapprove, had he thought the people would be receptive, is uncertain. It could well have been that Washington would've found Highness appropriate for the office of the president, but nonetheless rejected the much more grandiose Majesty that Adams later upgraded his proposal to. This may be the interpretation your history teacher chose.
In any event, working through James Madison (who likewise felt titles were harmless but against the mood of the nation), Washington prevented the Senate from addressing him as "His Highness the President of the United States, and Protector of their Liberties", and the issue has rested since.