The match is failing because you've put uppercase letters in your search string. From http://www.gnu.org/software/emacs/manual/html_node/emacs/Search-Case.html:
Searches in Emacs normally ignore the case of the text they are
searching through, if you specify the text in lower case. Thus, if you
specify searching for ‘foo’, then ‘Foo’ and ‘foo’ also match. Regexps,
and in particular character sets, behave likewise: ‘[ab]’ matches ‘a’
or ‘A’ or ‘b’ or ‘B’.
An upper-case letter anywhere in the incremental search string makes
the search case-sensitive. Thus, searching for ‘Foo’ does not find
‘foo’ or ‘FOO’. This applies to regular expression search as well as
to string search. The effect ceases if you delete the upper-case
letter from the search string.
Assuming the variable case-fold-search is set to t (the default), then if you use this search text:
""[a-z]+, [a-z]+"
and this replacement text:
"\&"
Then query-replace-regexp will convert this:
""Foo, Baz" <foobaz>@fizz.com>"
to this:
"""Foo, Baz"" <foobaz>@fizz.com>"