The point of quantile normalization is to make the signal distribution of the samples as close as possible, so you normalize the columns. Trying to normalize the rows would introduce a whole host of issues that could seriously muck with the results (e.g., since you look for differential expression across rows, how might normalizing each row affect that...).

Thank you very much for answering this question Devon, I have a follow-up question: Under what circumstances is it appropriate to do both log transformation and quantile normalization? Thank you in advance.