No clear way of finding two matches within a word within a series of words occurs to me (although Perl Mouse came up with one). I think the straightforward approach of splitting into words and then doing two checks per word is the way to go.

for (split ' ', $sequences) {
    if (/\Q$quart1/ and /\Q$quart2/) {
        print "Found both in $_\n";
    }
}
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The one issue to be concerned about is that the split could fill up memory. An alternative way to extract the words one at a time would be:

while (my ($word) = $sequences =~ /(\S+)/g) {
    if (index($word, $quart1) >= 0 and index($word, $quart2) >= 0) {
        print "Found both in $word\n";
    }
}
[download]