She told The Sun: "It was about half-an-hour to an hour into the flight I saw one. It was coming out of the back of the TV screen.
"Our food came out and I went to put the tray down on my lap. I saw what was maybe a flax seed - but it started moving - it was a bug.
"Once we got to the Airbnb that we were staying in, we went to sleep. My daughter had a few bites on her thighs but when she woke up she was covered, she had them all over. It was just so bad and awful, my daughter was bleeding."
The incident happened on Monday as the family were flying from their home in Canada to Slovakia, with a stop-over in London.
Ms Szilagyi said she was speaking out to shame BA into taking care of the problem, and questions whether she wants to fly with the airline on the return trip.
A BA spokesman said incidents of bedbugs on aircraft are very rare, but do happen "occasionally" and are dealt with seriously.
The spokesman said the airline has apologised to Ms Szilagyi, her fiance Eric Neilson and her daughter.