How can I update password using C#?

There are many many (mostly) minor mistakes in your code:

• use some kind of Id fields in your sql tables
• never do an update like you did (update the field WHERE this field is equals to...)
• create your own class and bind the query result to this class
• when a class implements IDisposable interface, always use the keyword 'using'
• never ever user string concatenation in sql queries!!! SQL INJECTION!!! always use parametrized sql queries

Here's a simple example for your form. Let's suppose your user.register table has the following columns: - Id - Username - Password

Now let's create your own class (maybe right under your button click event, so it can be private this time):

private class MyUser
{
    public int Id { get; set; }
    public string Username { get; set; }
    public string Password { get; set; }
}

Then your button click event should look like this:

private void Btnchange_Click(object sender, EventArgs e) {
if (!textnewpassword.Text.Trim().Equals(textconfirmpassword.Text.Trim()))
{
    throw new ArgumentException("New password and confirm password should be same!");
}

List<MyUser> myUsers = new List<MyUser>();

using (MySqlConnection con =
    new MySqlConnection(
        "server=localhost;user id=root;persistsecurityinfo=True;database=user;password=1234"))
{
    using (MySqlCommand cmd = new MySqlCommand("select * from user.register where Username=@user and Password=@pass", con))
    {
        cmd.Parameters.AddWithValue("@user", textusername.Text.Trim());
        cmd.Parameters.AddWithValue("@pass", textoldpassword.Text.Trim());

        if (cmd.Connection.State != ConnectionState.Open) cmd.Connection.Open();

        using (MySqlDataReader dr = cmd.ExecuteReader())
        {
            while (dr.Read())
            {
                myUsers.Add(new MyUser
                {
                    Id = (int)dr["Id"],
                    Username = dr["Username"].ToString(),
                    Password = dr["Password"].ToString()
                });
            }
        }

        if (cmd.Connection.State == ConnectionState.Open) cmd.Connection.Close();
    }

    if (!myUsers.Any())
    {
        throw new ArgumentException("No users found with the given username/password pair!");
    }

    if (myUsers.Count != 1)
    {
        throw new ArgumentException("More than 1 user has the same username and password in the database!");
    }

    MyUser user = myUsers.First();
    user.Password = textnewpassword.Text.Trim();

    using (MySqlCommand cmd = new MySqlCommand("update user.register set Password=@pass where Id=@id"))
    {
        cmd.Parameters.AddWithValue("@pass", user.Password);
        cmd.Parameters.AddWithValue("@id", user.Id);

        if (cmd.Connection.State != ConnectionState.Open) cmd.Connection.Open();
        cmd.ExecuteNonQuery();
        if (cmd.Connection.State == ConnectionState.Open) cmd.Connection.Close();
    }
} }

...and so on.