Assuming you only want sentence1 returned, you need to put your whitespace inside your wild card character, like

SELECT  ID,
        sentence  
FROM    texts
WHERE   sentence LIKE '% $query %'
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That way, you get your literal whitespace on either side of your tag. This suffers from limitation that it will miss for leading or trailing words and is unlikely to behave the way you intend around punctuation. You could do better with:

SELECT  ID,
        sentence  
FROM    texts
WHERE   sentence LIKE '% $query %'
   OR   sentence LIKE '$query %'
   OR   sentence LIKE '% $query'
   OR   sentence = '$query'
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But this still doesn't do a great job. If I were to do this, I would just do the simple query

SELECT  ID,
        sentence  
FROM    texts
WHERE   sentence LIKE '%$query%'
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and then filter the results against an appropriate regular expression, like /\b\Q$query\E\b/.

As a side note, you should consider using Placeholders_and_Bind_Values (see DBI) for database access. It protects you from all sorts of security exploits and handles messy escaping issues no trouble. I would actually do the above as

my $sth = $dbh->prepare(<<'EOSQL');
SELECT  ID,
        sentence  
FROM    texts
WHERE   sentence LIKE ? ESCAPE ?
EOSQL
$sth->execute("%$query%", '\\');
my $results = [];
while (my @row = $sth->fetchrow_array) {
    push @$results, \@row if $row[1] =~ /\b\Q$query\E\b/;
}
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