As a guess, as I'm not really familiar with Romeo and Juliet, 'that has brought blood to your face' would mean that someone was embarrassed. 'To go red' is an idiom/saying - to suffer embarrassment or shame.

It means to blush from embarrassment. I can't find your exact words in the play, though. What I find is: "Now comes the wanton blood up in your cheeks, They'll be in scarlet straight at any news." Since the word "wanton" means "related to sexual desire," the meaning of "blushing" is confirmed.