Add a text animator to your layer. Go to Range Selector>Advance> and change Units to Index and Based on to Lines.
Set Start to 1.
Now, if you want only second line to be moved to the right, set End to 2. If you want all your subsequent lines to go a bit to the right, set this End value to something high(the max number of lines you will ever have. Let's say 10).
Now go to Animator 1> Position and modify the first value of the position until you are happy with the result(this will vary based on your font and font size). You should do this with a text that starts with ".
Now alt+click this Animator1>Position and add this as an expression. You need to copy-paste the double quotes inside that expression because some fonts have different in/out quotes.
text.sourceText.substring(0,1) == '“' ? value : [0,0];