Random Generator from String Array

This is how you use random class in your case. Choose a random integer, where the highest number chosen by random is your total number of questions.

Random random = new Random();
int randomQuestion = random.nextInt(nrOfQuestions);

and you use this randomQuestion variable to access you question from the matrix: testBank[randomQuestion][0]

Cheers!

Shuffling and then shortening the array is probably the way to go. This is more easily done on lists however. In general collection classes should be preferred over arrays.

Most of the work in this answer is performed to convert the arrays to a list and back again.

private static String[][] randomize(String[][] testBank, int questions) {
    // convert top array to list of mutable size 
    List<String[]> testBankAsList = new ArrayList<>();
    for (String[] qoa: testBank) {
        testBankAsList.add(qoa);
    }

    // randomize questions
    Collections.shuffle(testBankAsList, new SecureRandom());

    // remove the tail
    testBankAsList.subList(questions, testBankAsList.size()).clear();

    // convert back into array
    String[][] shorterRandomTestBank = testBankAsList.toArray(new String[testBankAsList.size()][]);

    return shorterRandomTestBank;
}

Where questions should be set to 2 of course. I left out all the parameter checking.

Note that Arrays.toList cannot be used as it simply creates a list over the backing array, and therefore the clear() operation will fail.

Example usage:

public static void main(String[] args) {
    String testBank[][] = {{"What color is grass?","A. Green","B. Red","C. Pink","A"},
            {"Whats the first month called?","A. December","B. January","C. March","B"},
            {"What shape is a soccer ball?","A. square","B. flat","C. round","C"}};
    int questions = 2;

    String[][] shorterRandomTestBank = randomize(testBank, questions);

    // iterate of the returned questions
    for (String[] qoa : shorterRandomTestBank) {
        // prints the question
        System.out.println(qoa[0]);

        // prints the answer
        System.out.println(qoa[qoa.length - 1]);
    }
}

Sorry, I'll let you do the Swing / GUI programming yourself. If you get into trouble with that ask a separate question.

Here is an answer that handles the possibility that the random generator generates the same question twice. With the code below you will only ever get 1 question asked. Also I have commented each step to give you an idea of the logic. One small point if you had a 1D array of type int, then you would run into problem during the check. This is because 1D arrays of type int set the default value to 0. Using Integer or an ArrayList eliminates this problem because default value is null and not 0.

I have also included the loop and JOptionPane code required to display the randomly chosen question. It is up to you to decide what you want to do with the responses.

public static void main(String args[]) {

    // We are setting this final variable because we don't want to hard code numbers into loops etc
    final int NUMBER_OF_QUESTIONS_TO_TAKE = 2;

    String testBank[][] = {{"What color is grass?", "A. Green", "B. Red", "C. Pink", "A"},
            {"Whats the first month called?", "A. December", "B. January", "C. March", "B"},
            {"What shape is a soccer ball?", "A. square", "B. flat", "C. round", "C"}};

    // Initialise the array of questions that have been randomly chosen.
    String finalArrayOfQuestions[] = new String[NUMBER_OF_QUESTIONS_TO_TAKE];

    // ArrayList to store the index number of a question so we can check later if it has been already used by number generator
    ArrayList<Integer> alreadyChosenList = new ArrayList<Integer>();

    // boolean that we will use for whether or not a question has already been selected
    boolean alreadyChosen;

    // The column number that the random number generator generates, which is then used to extract the String question
    int rowToUse;

    // A for loop is used to loop through the process, depending on how many questions you want to take.
    for (int i = 0; i < NUMBER_OF_QUESTIONS_TO_TAKE; i++) {

        // Generate a random number, repeat the process (do/while) until a random number has been generated that hasnt been generated before
        do {
            // Generate a random number within the range
            Random random = new Random();
            rowToUse = random.nextInt(testBank.length);

            //check not already been picked
            alreadyChosen = alreadyChosen(rowToUse, alreadyChosenList);
        } while (alreadyChosen);


        // Get String representation of question chosen at random
        String questionChosen = testBank[rowToUse][0];

        // Add this String to finalListOfQuestions
        finalArrayOfQuestions[i] = questionChosen;

        // adds to list of questions already chosen. Makes sure you don't take same question twice.
        //alreadyChosenList[i] = alreadyChosenList[rowToUse];
        alreadyChosenList.add(rowToUse);
    }

    for (String questions : finalArrayOfQuestions) {
        String response = JOptionPane.showInputDialog(questions);

        /*
        The response is the answer that the user types in. Here you can check it against the arrays you have
        Or if you dont want the user to input a response use:
         JOptionPane.showMessageDialog(null, questions);
         */

    }
}


/*
Method takes row index to use and the ArrayList of row indexes already been used and returns true or false depending if the current one is in the arraylist
 */
private static boolean alreadyChosen(int rowToUse, ArrayList<Integer> alreadyChosenList) {

    for (int indexToCheck : alreadyChosenList) {
        if (indexToCheck == rowToUse) {
            return true;
        }
    }
    return false;
}