Do you put quotation marks in front of the word she told me?

Yes, if your using a quote from her you do use quotation marks. Like this-I'll quote you. The question was "Do you put quotation marks in front of the Word she told me?"

The phrase she told me is what in language we refer to a dialogue tag. It is introductory to a quote. For instance, in the example below:

She told me, "You are wasting your time waiting for Bob to propose to you."

Dialogue tags need not precede the quote. They can appear between any clauses in the quoted sentence. For instance:

"You," She told me, "are wasting your time waiting for Bob to propose to you."

"You are wasting your time," She told me,"waiting for Bob to propose to you."

"You are wasting your time waiting for Bob,"She told me, "to propose to you."

"You are wasting your time waiting for Bob to propose," She told me, "to you."

You'll notice that each placement of the dialogue tag tends to emphasize a different part of the quoted sentence, and can become awkward.