I've been using PhpStorm since v5, but have not found a way to address this.
However, this behaviour can vary, depending on how you're getting to the quotes.
- If you're typing
<div class=
and it's automatically filling in the""
for you, you can remove that by [unsetting "Automatically add quotes for attribute values"]. - If you're using tab-completion, like
<div cla[TAB]
, then there doesn't seem to be much you can do, except press backspace after tab. I haven't - You can set up your own Live Template
<div class='$END$'></div>
to give you your desired format.
There may be a plugin or two that can change / override PhpStorm's default behaviour, but I haven't looked too much.