Old phpBB posts use different [quote] BBCode markup that doesn't work anymore

I restored a ten-year-old phpBB forum, and after upgrading it to 3.1.x, I noticed that the quote BBCode is not working correctly (but I don’t know if it worked in the old version).

If no username/URL is specified, everything’s fine:

[quote]…[/quote]

But if the username/URL is specified, the format in old posts looks like this:

[quote=John]…[/quote]

while phpBB seems to require this format (and this is also what the post editor generates):

[quote="John"]…[/quote]

So the quotes around the name/URL are missing.

How could I convert this BBCode in old posts? Or is it possible to make the version without quotation marks work in addition to the new one?