Sure. The Java code describes the computation you want done, just like the math-like definition of Triangle.
Here is a complete program for testing this method. The value for N is hard-coded. You might wish to improve the program so that N is entered by the user.
class TriangleCalc
{
  int Triangle( int N )
  {
    if ( N == 1 )
      return 1;
    else
      return N + Triangle( N-1 );
  }
}
class TriangleTester
{
  public static void main ( String[] args)
  {
    TriangleCalc tri = new TriangleCalc();
    int result = tri.Triangle( 4 );
    System.out.println("Triangle(4) is " + result );
  }
}
Here is the output of the program:
C:\>java TriangleTester Triangle(4) is 10 C:\>
Of course, it would be worth while at this point to copy this program to a file and run it.
What is Triangle(500)?