If I'm understanding you want to remove the double quote from the left side of the entry in Name_Last column, is this correct?
If so them you can try this.
$stripname = $db->sql_query("UPDATE ".$prefix."$table_name SET Name_Last = TRIM(LEADING '\"' From Name_Last");
Btw I don't think you can use a PHP function in a SQL query, I could be wrong though.