Are they "straight" quotes, or are they “directional” quotes?
If they are straight quotes, one method I can think of would require two
steps (unless you want the quote marks to also be bold):
1. Set Find:"*" Enable Use Wildcard. Press Ctrl+B twice so that Not Bold
is set as the Find formatting. Click in the Replace field, and press Ctrl+B
once so that Bold is set at the replace formatting; make sure that the
Replace field is empty. Click Replace All.
2. To unbold the quotes, Set Find:" only, set the formatting to Bold; clear
the Use Wildcards option, and set the formatting of the [still empty]
Replace field to Not Bold. Click Replace All.
Obviously, if you otherwise have any quotes that you want kept bold, step 2
isn't what you want.
If they are directional quotes that you have, you should type “*” into your
document, copy it to the clipboard, then paste it into the Find field,
rather than using "*".