Let DBD::ODBC do the quote escaping, it knows what kind of escapes to use. This can be done with $dbh->quote() as others have said, but it can also be done with placeholders and placeholders have many other benefits. Read the DBI docs about placeholders, here's an example:

  my $sth = $dbh->prepare(q{
      INSERT INTO TRANS_TABLE 
                  (Id, TransString, PoundDef, Filename)
           VALUES( ?, ? , ?, ?)
  });
  $sth->execute(
      0, 
      q{Attempt' to connect to %U?},
      q{qtn_ics_ap_conne ct_text}, 
      q{agentdialog_00.loc}
  );
[download]