here

1. int [][] array1 = { //all possible combinations
2.   {
3.     1, 2, 3, 4
4.   }
5.   ,
6.   {
7.     0, 1, 2, 4
8.   }
9.   ,
10.   {
11.     3, 4, 5, 6
12.   }
13.   ,
14.   {
15.     2, 3, 9, 3
16.   }
17.   ,
18.   {
19.     3, 4, 6, 8
20.   }
21.   ,
22.   {
23.     3, 4, 12, 13
24.   }
25. };
26. int [][] array2 = {//all combinations for this player
27.   {
28.     1, 2, 3, 4
29.   }
30.   ,
31.   {
32.     0, 1, 2, 4
33.   }
34.   ,
35.   {
36.     3, 4, 5, 6
37.   }
38.   ,
39.   {
40.     2, 3, 10, 11
41.   }
42.   ,
43.   {
44.     3, 4, 12, 13
45.   }
46. };
47. int [][]array3 = new int [6][4];
48. void setup() {
49.   int count=0;
50.   for (int i=0; i<array1.length;i++) {
51.     for (int j=0;j<array2.length;j++) {    
52.       if (Arrays.equals(array1[i], array2[j])) {      
53.         array3[count]=array2[j];
54.         // println(array2[j]);
55.         // println("next");
56.         count++;
57.       } // if
58.     } // for
59.   } // for
60.   //
61.   // --------------------------------
62.   println("result");
63.   for (int i=0; i<count;i++) {
64.     println (i+": ");
65.     println(array3[i]);
66.   }
67. } // func
68. //
69. void draw() {
70.   //
71. }
72. //

If you have remaining empty slots, you can truncate the array.
You used Arrays.equals() (good idea), you can also use Arrays.copyOf() with the value of count to make a new array of the right size.
You have to know that when you declare int[6][4], you actually create an array of six entries, each of these entries being filled by an array of 4 entries. In other words, a two dimension array in Java is just an array of arrays (and the contained arrays can be of various sizes!).
Actually, if you declare your array like:
int [][]array3 = new int [6][];
it will work too, and will take less memory (with lot of rows): you create rows, but actually you discard them in the inner assignment.

Alternatively, you can use an ArrayList to build the result. If needed, it is easy to convert it to an array afterward.

That's good to hear!