When Juliet find out Romeo's last name?

Juliet discovers that Romeo is a Montague in act 1, scene 5. She has just had a conversation with him and is infatuated. After their talk, she asks the nurse to find out his name and says something remarkably prophetic:

Go ask his name: if he be married.My grave is like to be my wedding bed.

Her words foreshadow the tragic events which are to follow later in the play.
Juliet is shocked when the nurse tells her that Romeo is a Montague, and she cries out in despair:

My only love sprung from my only hate!Too early seen unknown, and known too late!Prodigious birth of love it is to me,That I must love a loathed enemy.

Her statement refers to the age-old feud between the Capulets and the Montagues. The two families are lifelong enemies, and Juliet realizes that she has fallen in love with a member of a despised family. She grasps what the situation implies and the complications thereof. In her distress, she remarks that it is a pity that she has fallen for Romeo before actually knowing who he is. Her love will now be tarnished by the insidious presence of the two families' hatred for each other. It is in this sense, then, that her newfound passion has had a momentous birth.
Juliet's remark also makes it apparent that although she realizes that her love for Romeo may be an immense mistake, she will still love him. She states that she "must love a loathed enemy." It is as if she is destined to love him and has no choice in the matter.
It is this belief which informs Juliet's actions from this moment on and eventually culminates in great tragedy.