Make a careful consideration of what should happen with all the cases.
(using [] as quotes for readability)
You've said what you do for ["Hello"] but what do you do for [I said "Hello", punk]? Do you still strip the user-input quotes, or do you remove them? Or maybe go one step further and substitute single quotes for double quotes...but then you'd have to consider the cases where the user input contains both single and double quotes! What about when the user puts in "grammatically wrong" text like ["Hello] (no closing quote!).
Best way to ensure you do it right is to make a test case for each edge case you can think of and make sure your proposed solution actually does what is expected.
If the actually reason for this requirement is [I am taking a user's input at some point and then re-displaying it to them at another point, and don't want to show them stupid looking data like [""data""]], you'll probably just want to only remove double quotes from the start or end of the input string, because stripping them from the middle screws with the user's intended data.
If your goal is merely to clearly distinguish between their input and text that they didn't input, consider using other means of highlighting that instead of quotation marks. So instead of [you entered "data"], you could display [you entered data] or [you entered: data] which avoid this problem altogether.