I don't think we will ever know for sure. The royal family is notoriously tight lipped about all private matters and they won't say anything that will present the Queen in a bad light.

The origin of this representation of the Queen's education appears to be British historian David Starkey who described her as poorly learned and said she lacked cultural refinement and intellectual curiosity in the 2007 television series Monarchy and also an interview with The Guardian. However, this was met with quick rebuttal from several people like the Royal biographer Penny Junior who said: "The Queen is certainly cultured even if not that moved by the arts. The Prince of Wales has a great sense of history and a lot of that comes from his mother."

You can read the interview with David Starkey here: https://www.theguardian.com/uk/2007/dec/22/monarchy.television

In my opinion his opinion seems to be mostly based on her cool reaction to him (which he blames on her being uncomfortable with intellectual people), her reaction to an art exhibition he put on about Elizabeth I (indifference which he blames on her not being interested in her ancestors beyond her father and grandfather) and an analysis of all her Christmas speeches in which she also mostly references her father and grandfather when talking about the royal family. At least that's how the interview presents it.

Whether his interpretation is correct or if it is the result of his hurt ego, who knows.

I am actually also wondering this; in general, how accurate is the show, and are there any known factual inaccuracies?