The closeness of Juliet and the Nurse can be seen in the scene as said above where the Nurse recounts their bond since her childhood. However, I think the most compelling actions are when the Nurse seeks out Romeo on Juliet's behalf. Juliet's mother is concerned with status and money when it comes to Juliet's marriage, but the Nurse wants Juliet's happiness. She sets Juliet and Romeo's marriage up:
"Then hie you hence to Friar Laurence' cell;
There stays a husband to make you a wife:..." (II.v.)
It is also evident in act 4 of the play when the Nurse is distraught over Juliet's (fake) death saying
"Mistress! what, mistress! Juliet! fast, I warrant her, she: Why, lamb! why, lady! fie, you slug-a-bed!" (IV.iii.).
The Nurse does not take the news well. She is more than able to set Juliet and Romeo up to elope together knowing that although her pain will be great in losing Juliet, Juliet will be happy. When she thinks that Juliet is dead, she repeats dramatic language like "woeful" and "lamentable."
The easiest place to find this quote from Shakespeare's Romeo and Juliet is in Act One, Scene 3. Here, Lady Capulet asks the nurse to call for Juliet, so that she might speak with her about getting married. The nurse recalls moments from when Juliet was a child and she actually took care of and "nursed" Juliet when she was a "babe", even including how her [the nurse's] late husband interacted and helped raise the child. But an interesting quote to use would be when the nurse calls for Juliet:
Nurse: God forbid! Where's this girl? What, Juliet! Juliet: How now! Who calls? Nurse: Your mother.
The answer really is that the nurse is calling Juliet for Juliet's mother. But Shakespeare establishes the clear connection of Nurse to Mother when she responds with this interesting answer.