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Is the sentence- Mary told me "What a nice Idea" - correct.
As tdol adds, it's best to use 'say' when quoting speech because the words represent the spoken word, an utterance. With 'tell' a similar sentence is possible but without the quotation marks,
Mary told me (that) it was a nice idea.
The underlined portion represents the information Mary gave you. She uttered similar words, but not those words exactly.
Although both of these words are used in both direct and indirect speech, say is more usual than tell in direct speech. This is because say can refer to any sort of speech, while tell always means "instruct" or "inform".
"Put the money in the slot", he said.
"Put the money in the slot", he told me.
"You have to leave now", she said.
"You have to leave now", she told me.
We said, "Happy birthday!" (NOT We told him, "Happy birthday").
"Where do you live?" he said. (NOT "Where do you live?" he told me).
The verb tell is usually followed by an object, that is, we usually say who is told.
I told her that she was wrong.
They told us that we could keep the objects.
Say is more commonly used without an object, but when it does have one, we use the word "to".
I said that she was wrong. (NOT I said her that …).
I said to her that she was wrong.
When tell has the meaning of "instruct", it can be followed by an object and an infinitive.
She told me to leave.
They told us to keep the objects.
We cannot use say in this way. Tell is used (with or without a personal object) in some common expressions:
Tell a story (He told (us) a frightening story).
Tell a joke (I'll tell (you) a joke I heard today).
Tell a lie (She told (me) a lie).
Tell the truth (Next time, she will tell (me) the truth).
Tell the difference (Can you tell (me) the difference between a frog and a toad?).
Tell the time (I can't tell (you) the time because it's too dark).
Say/Tell 
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