For literal strings, I like index,

my ($loc, %location) = -1;
for ($quart1, $quart2) {  # or better, @quart
    while (-1 != $loc = index $sequences, $_, $loc + 1) {
        push @{$location{$_}}, $loc;
    }
}
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There is some tricky fenceposting there with $loc. The %location hash will wind up containing all the locations in the string of each $quart which is found at least once. You will be able to test if a string is present at all with exists $location{$str}.

This code doesn't care at all about overlap, will easily find the presence of both in those cases.