import java.time.LocalDate;
import java.time.Month;

public class Birthday{
   public static void main(String args[]) {

    // declare variables for birthday
    int birthDate = 23;
    Month birthMonth = Month.SEPTEMBER;

    // get current date
    LocalDate currentDate = LocalDate.now();
    System.out.println("Todays Date: " + currentDate);

    // get current date and month
    int date = currentDate.getDayOfMonth();
    Month month = currentDate.getMonth();

    if(date == birthDate && month == birthMonth) {
      System.out.println("HAPPY BIRTHDAY TO YOU !!");
    }
    else {
      System.out.println("Today is not my birthday.");
    }
   }
}

Your code runs successfully. See the code run live at IdeOne.com.

Java 8+

Since you gave few details, I will guess that your problem is due to using a very old version of Java.

The java.time classes are built into Java 8 and later, and Android 26+.

• If using Java 6 or 7, get most of the java.time functionality by adding the ThreeTen-Backport library to your Java project.
• If using earlier Android, the latest tooling provides most of the java.time functionality via “API desugaring”.

I will go on to mention a couple of other issues in your code.

Enum#equals

You should replace the use of == with object references such as Month objects. While that technically works with enum objects, using == on objects is a bad habit, with no benefit gained over calling Enum#equals.

if( date == birthDate && month.equals( birthMonth ) ) {

MonthDay

Make a briefer version using MonthDay.

MonthDay currentMonthDay = MonthDay.now() ;
MonthDay birthMonthDay = MonthDay.of( Month.SEPTEMBER , 23 ) ;
if( currentMonthDay.equals( birthMonthDay ) ) {…}

Or a one-liner:

if( MonthDay.now().equals( MonthDay.of( Month.SEPTEMBER , 23 ) ) ) { … }