ShowStringCharacters

is an option for Cell that specifies whether to display " when a string is entered.

Details

ShowStringCharacters is typically set to False for output cells and True for input cells.
ShowStringCharacters is usually set at the level of styles or notebooks rather than at the level of individual cells.

By default, quotation marks are shown around strings in input cells, but not in output cells:

Create a cell in "Output" style but with quotation marks showing:

Show a grid of characters without quotation marks:

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