Obviously, you already know about SQL injection. So let's forget about that. Ideally, please do defend against it, it's easy enough that it's not worth doing it any other way. Code that is not public facing now, might be next week.

Some other things:

You use fetchall instead of fetchone. fetchall has to load all rows into memory (I assume), which is pretty expensive. Don't use it.

Also, you use SELECT *, which needs to load all columns (of the relevant rows) into memory. Again, expensive.

def check_word(self, word_name):
    self.c.execute('SELECT {c2n} FROM {tn} WHERE {c1n} = """{wn}"""'.format(tn=self.table1, c2n=self.column2, c1n=self.column1, word=word_name))
    data = self.c.fetchone()
    if data is not None:
        new_freq = data[0] + 1
        self.c.execute(
            'UPDATE {tn} SET {c2n}={en} WHERE {c1n}="""{word}"""'.format(tn=self.table1, c2n=self.column2,
                                                                     en=new_freq, c1n=self.column1,
                                                                     word=word_name))
        return True
    return False

Also, there is a bit of a trouble with race-conditions here, but that's probably easily solvable using transactions and locking. Or, better, use 1 query (which is also faster!)

def check_word(self, word_name):
    query = 'UPDATE {table_name} SET {counter} = {counter} + 1 WHERE {word_column} = ?'.format(
        table_name=self.table1,
        counter=self.column2,
        word_column=self.column1,
    )
    self.c.execute(query, word_name)
    return self.c.row_count > 0

(Here, I use the row_count attribute of a cursor which returns the number of rows changed in the last query: https://docs.python.org/2/library/sqlite3.html#sqlite3.Cursor.rowcount).

I tried to use more sensible names for the columns, and also used string-interpolation for the values that are hopefully not under user control (and can't be done using parametrized queries anyhow), while using paramatrisation for the word_name which is more likely to change.

Obviously, you already know about SQL injection. So let's forget about that. Ideally, please do defend against it, it's easy enough that it's not worth doing it any other way. Code that is not public facing now, might be next week.

Some other things:

You use fetchall instead of fetchone. fetchall has to load all rows into memory (I assume), which is pretty expensive. Don't use it.

Also, you use SELECT *, which needs to load all columns (of the relevant rows) into memory. Again, expensive.

def check_word(self, word_name):
    self.c.execute('SELECT {c2n} FROM {tn} WHERE {c1n} = """{wn}"""'.format(tn=self.table1, c2n=self.column2, c1n=self.column1, word=word_name))
    data = self.c.fetchone()
    if data is not None:
        new_freq = data[0] + 1
        self.c.execute(
            'UPDATE {tn} SET {c2n}={en} WHERE {c1n}="""{word}"""'.format(tn=self.table1, c2n=self.column2,
                                                                     en=new_freq, c1n=self.column1,
                                                                     word=word_name))
        return True
    return False

Also, there is a bit of a trouble with race-conditions here, but that's probably easily solvable using transactions and locking. Or, better, use 1 query (which is also faster!)

def check_word(self, word_name):
    query = 'UPDATE {table_name} SET {counter} = {counter} + 1 WHERE {word_column} = ?'.format(
        table_name=self.table1,
        counter=self.column2,
        word_column=self.column1,
    )
    self.c.execute(query, word_name)
    return self.c.rowcount > 0

(Here, I use the rowcount attribute of a cursor which returns the number of rows changed in the last query: https://docs.python.org/2/library/sqlite3.html#sqlite3.Cursor.rowcount).

I tried to use more sensible names for the columns, and also used string-interpolation for the values that are hopefully not under user control (and can't be done using parametrized queries anyhow), while using paramatrisation for the word_name which is more likely to change.