The most obvious fate quote is the opening prologue. It summarizes most of the play in its 14 lines. That's probably cheating though.
Romeo, on his way over to the Capulet ball, tells the reader that he has some misgivings about events that will be set in motion that night.
"I fear, too early: for my mind misgives
Some consequence yet hanging in the stars
Shall bitterly begin his fearful date
With this night's revels and expire the term
Of a despised life closed in my breast
By some vile forfeit of untimely death."
At the ball, just before Romeo and Juliet meet, Juliet asks her nurse to find out who that guy (Romeo) is. She makes a prediction then that links her wedding and her grave.
"Go ask his name: if he be married,
My grave is like to be my wedding bed."
Mercutio screams out the following: "A plague a’ both houses!" Yep, he's right. There is a plague on both houses. There has been, and it will get worse before the play is over.
Toward the close of the play Friar Laurence has a great fate quote in which he talks about a higher power of sorts that ruined his plans.
"A greater power than we can contradict
Hath thwarted our intents."