Assuming all you want to know is whether students scored better after instruction than before, then yes, you can use MANOVA to analyze the data. However, an easier method would be to use repeated measures ANOVA. Repeated measures ANOVA is a special case of MANOVA, so assuming you meet the assumptions of repeated measures ANOVA, you would be doing the same thing as MANOVA. The only caveat is that repeated measured ANOVA assumes your data is spherical (while the more general MANOVA does not). If it is not, then your results will likely be invalid.
While there are lots of tutorials out there on doing repeated measures ANOVA, most are used to find main effects and interactions in a classic ANOVA style. You have a specific question in mind, so it would be easiest to simply test that specific question: "Does the pretest score differ from the post test score?" This is equivalent to asking if $(Q21 + Q22) - (Q11 + Q12) > 0$
In R this is easily done by a simple linear model, and testing intercept.
dat$quiz.diff <- (dat$Q21 + dat$Q22) - (dat$Q11 + dat$Q12)
Other questions would likely be assessed using different contrasts between the variables. Note that while you are not using any special repeated measures functions, this is still a repeated measures ANOVA. You are simply doing some of the math yourself in making the contrast.
Edit: I saw your response below where you say you want to know which questions caused the increase in quiz scores. I am still unclear as to exactly what the rest of your data looks like, so I'm just going to go on what you provided. If one question contributed more to the quiz score increase than the other, that means that the change in question 1 is significantly greater or less than the change in question 2. Or $(Q21 - Q11) > (Q22 - Q12)$ which is equivalent to $(Q21 - Q11) - (Q22 - Q12) > 0$. So you would make a new variable with that contrast, and create a new linear model with that variable, testing the intercept of the model.