Both are grammatically correct; however my first impulse is to go with "Quote of the Day" rather than with "Quote for the Day." You usually see "Quote of the Day" in publications. However there is no obvious difference and meaning and you could use both interchangeably.
Quote of the Day May 12 We are told that talent creates its own opportunities. But it sometimes seems that intense desire creates not only its own opportunities but its own talents.
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