I read the first line of cchampion's reply but not the rest, so I thought I'd take a stab myself.

Like the OP, I presume the variable $dbh, the array @userdata and the subroutine error() to be predefined. However, it appears that the OP is using this subroutine to set a global variable called $sth from which results are later retrieved. I find this upsetting so I'm going to have the subroutine termgrades() return it instead, and return undef in case of errors. I also changed the messages so that an prepare error would not return the same message as an execute error.

my %quarternames = (1 => 'first',  2 => 'second', 3 => 'third', 4 => '
+fourth');

my %tablenames = map { $_ => "$quarternames{$_}quarter" } 
                      keys %quarternames;

my %queries = map { $_ => " SELECT * FROM $tablenames{$_} WHERE 
                            userid = ? AND period = ? AND gradetype = 
+? " 
                  } keys %tablenames;

sub termgrades {
    my $period  = shift;
    my $quarter = shift;
    my $sth;
 
    return undef unless exists $queries{$quarter};
    $sth = $dbh->prepare_cached($queries{$quarter})
           or do { error ("2", "Could not prepare query for $quarterna
+mes{$quarter} quarter for $userdata[0]: $dbh->errstr");
                   return undef;
    };

    $sth->execute($userdata[0], $period, 2)
         or error("2", "Could not execute query for $quarternames{$qua
+rter} quarter for $userdata[0]: $sth->errstr");
    return $sth;
}
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Update: Made hash initialization slightly more clever.