ID does not have a proper hanging control.

In regular text you can combine a left indent with a first line negative indent to simulate this –a solution valid just for the paragraph opening. The other resource is to activate optical adjust for margins –again, a very limited procedure for achieve hanging aligment, useful most for quotes and capitular swashes.

In a table, the same limitations from regular text are also present. You can just simulate the hanging effect placing the brackets in a thin first column and the remaining text is the next one. A sample:

Example above shows optical margins adjust working in table. Table below pretends a hanging aigment with additional column at left.

A hack:
Insert a space before the quote mark, and negative kern.

I'd probably use the first method that Freiberger suggested -- left indent with negative first line indent. I'd create a style sheet for that formatting and apply it to all paragraphs that begin with a bracket.

If you also have some paragraphs in that table that don't start with a bracket, I'd create a separate style sheet for them that was identical except it didn't have the 1st line negative indent.

If the bracket is not on the paragraph's first line, you'll need to put a hard break at the end of the line above it, so that InDesign views the bracketed line as the start of a new paragraph.

Thanks for the helpful responses. I went for adding an extra column in the end.