"With the sleep of dreams comes nightmares" - Is this a Shakespeare quote and if so, which play is it from (the Act and Scene). Thanks.

This particular quote is not a direct quote from
Shakespeare.  The word "nightmare" only appears once in Shakespeare's plays and it is
not in a context anything like this quote.  (This is according to a couple of sites
where you can search all of Shakespeare's works at once.)

I
think that you may be thinking of a famous line that is from Shakespeare and has a
somewhat similar meaning.  It is from Hamlet's "to be or not to be" soliloquy in Act
III, Scene 1 of Hamlet.  In that passage, Hamlet
says

To sleep-
perchance to dream: ay, there's the rub!
For in that sleep of death what
dreams may come
When we have shuffled off this mortal
coil

That sort of means the
same thing -- if you dream, you may have nightmares.