The easiest way to format this in Indesign is to use the Indent to Here function. Place the cursor at the point to the right of the quote mark then go to Type > Insert Special Character > Other > Indent to Here:
If this function doesn’t work correctly/behave, it could be that there is unruly formatting in your text such as tabs or hard returns (especially if the text was copied from the web.)
Go to the Type menu > Show Invisible Characters to show spaces, tabs, returns and other formatting:
The red marks show formatting. Dots are spaces, the sideways ‘L’ at the end of the line are Soft Returns, whereas the Paragraph signs are Hard Returns (as you can see, the hard return causes too much spacing after the line.)
A soft return (Shift+Enter) is simply a line break whereas a hard return (just “Enter”) indicates a new paragraph, which becomes more of an issues once you begin to use more complex text and space formatting (indents, space above and below, etc.), but it is also an issues using the “Indent to Here” function because Indesign sees the hard return as the end of the paragraph that needs to be indented.