H2 - How to create a database trigger that log a row change to another table?

Declare this trigger:

CREATE TRIGGER my_trigger
BEFORE UPDATE
ON my_table
FOR EACH ROW
CALL "com.example.MyTrigger"

Implementing the trigger with Java/JDBC:

public class MyTrigger implements Trigger {

    @Override
    public void init(Connection conn, String schemaName, 
                     String triggerName, String tableName, boolean before, int type)
    throws SQLException {}

    @Override
    public void fire(Connection conn, Object[] oldRow, Object[] newRow)
    throws SQLException {
        try (PreparedStatement stmt = conn.prepareStatement(
            "INSERT INTO log (field1, field2, ...) " +
            "VALUES (?, ?, ...)")
        ) {
            stmt.setObject(1, newRow[0]);
            stmt.setObject(2, newRow[1]);
            ...

            stmt.executeUpdate();
        }
    }

    @Override
    public void close() throws SQLException {}

    @Override
    public void remove() throws SQLException {}
}

Implementing the trigger with jOOQ:

Since you added the jOOQ tag to the question, I suspect this alternative might be relevant, too. You can of course use jOOQ inside of an H2 trigger:

    @Override
    public void fire(Connection conn, Object[] oldRow, Object[] newRow)
    throws SQLException {
        DSL.using(conn)
           .insertInto(LOG, LOG.FIELD1, LOG.FIELD2, ...)
           .values(LOG.FIELD1.getDataType().convert(newRow[0]), 
                   LOG.FIELD2.getDataType().convert(newRow[1]), ...)
           .execute();
    }

Shorter version of Lukas Eder's answer:

CREATE TRIGGER my_trigger
BEFORE UPDATE
ON my_table
FOR EACH ROW
CALL "com.example.MyTrigger"

public class MyTrigger extends TriggerAdapter {

    @Override
    public void fire(Connection conn, ResultSet oldRow, ResultSet newRow) throws SQLException {
        // mannipulate the rows here by using the methods on the oldRow and newRow objects
    }
}