Drop caps work per line. So if you have a quote mark that drops 2 or more lines the text gets indented.
A better way to do this is create a character style for quotation marks.
Your body text needs to have an aboslute leading value - that is don't set the LEading to Auto.
If it's 10 pt type, then perhaps use a 16pt leading for the body text.
Character style settings for quotes mark
Now with the Quotation mark increase the font size, and use a Negative value for the Baseline Shift
But the problem with this now is that if the quote mark is at the start of a text frame the text drops way down.
So for Text Frames you need to go to the Object>Text Frame Objects and go to the Baseline tab in the options, set the first baseline to Leading.
Make sure you use styles for the Quotes, styles for text frames, styles for your body text etc (this will speed up the process)