As Jefromi has mentioned in his comments, I don't think you need to worry about what format the questions come in. In other words, who cares if the questions come on cards? Almost every modern board game involves cards but those don't seem boring for their use of a card mechanism.
The really important thing for maintaining interest is what happens around those cards. What makes the player have to draw or use those cards, what they have to do to utilize those cards, what they can use cards for?
I think it would be really cool to have a "cooperative" board game, rather than the standard competitive ones. I think a very stripped down and simplified Battlestar Galactica board game would be really cool. In that game, every turn the players have to turn over a "Crisis" card. The Crisis card is some kind of challenge or test that all of the players must help out in to complete. These cards could either make something good happen if the players succeed, or have a setback if the players fail. These tests could be some kind of trivia test or a math question or any kind of educational quiz. Each card could have two conditions - if the current player can pass the test by themselves (for a bigger bonus), or if they need to ask for help from their teammates. This would encourage cooperation, knowledge sharing, but also a focus on each player being accountable without being a huge penalty if they don't know all the questions.
A key piece to keep people involved is to make sure that the entire game isn't just answering questions. Even a trivia game such as Jeopardy has a betting mechanic in Double Jeopardy's and Final Jeopardy - this keeps people interested. Make sure that when you construct the game, there are interesting things the players can do to advance the game that aren't solely based on the material being taught. Again, games like Battlestar Galatica or Red November may be good templates for making decisions and moving around, in addition to some mechanism that is out of their control (ie. the trivia question cards).