You can use this:
$result = preg_replace('~\G(?!\A)(?>(\[quote\b[^]]*](?>[^[]+|\[(?!/?quote)|(?1))*\[/quote])|(?<!\[)(?>[^[]+|\[(?!/?quote))+\K)|\[quote\b[^]]*]\K~', '', $text);
details:
\G(?!\A) # contiguous to a precedent match
(?> ## content inside "quote" tags at level 0
( ## nested "quote" tags (group 1)
\[quote\b[^]]*]
(?> ## content inside "quote" tags at any level
[^[]+
| # OR
\[(?!/?quote)
| # OR
(?1) # repeat the capture group 1 (recursive)
)*
\[/quote]
)
|
(?<!\[) # not preceded by an opening square bracket
(?> ## content that is not a quote tag
[^[]+ # all that is not a [
| # OR
\[(?!/?quote) # a [ not followed by "quote" or "/quote"
)+\K # repeat 1 or more and reset the match
)
| # OR
\[quote\b[^]]*]\K # "quote" tag at level 0