cannot convert int to String, / cannot convert String to int [][]

 String [][] slsTablePP = slsData[3][Months]; 

Months is not an integer. You cannot directly convert an int array to a String array. To convert an integer array to a string array take a look at Converting an int array to a String array

String[]Months= new String [11];

The size of the above array should be 12.

slsData[3][Months]

Months should be an integer here, to access the nth element of the 3rd array in slsData. But the type of Months is String[], so that doesn't make sense. Maybe what you want to pass is the length of the Months array. In that case, you would use slsData[3][Months.length].

But since slsData is an array of array of ints, slsData[3][12] that would be an int. So you can't assign it to a variable of type String[][].

Note that you should respect Java naming conventions. Variables should have meaningful names, made of words (what do slsData and slsTablePP mean?), and should start with a lowercase letter (i.e. months, not Months).

Months is simply the name of your String array. In order to access individual values within the array you have to pass it the index of the value you want using the array operator [].

For instance:

String[] stringArray = new String[4];
String fourthString = stringArray[3];

or

String[] stringArray = new String[someList.size()]
for (int i=0; i<stringArray.length; i++) {
   stringArray[i] = someList.get(i);
   //some other stuff maybe
}

Obviously, this line is wrong:

String [][] slsTablePP = slsData[3][Months];

Arrays are typed and you cannot magically convert int[][] to String[][], furthermore, in Java, arrays are always indexed by int. This slsData[3][Months] where Months is of type String[][] has no meaning.
Note, in some language, like Python, you have associative arrays (Map and derived classes in Java) that can have an array syntax with string as index, e.g.

# python syntax
salesByMonth = {}
salesByMonth["jan"] = 2000;
salesByMonth["feb"] = 500; 

You have another issue when you declare the Months array. In

String[] Months= new String [11]; // <= should be 12 !

The value [11] is the size of the array, not the last index, hence it should be 12. In Java arrays are 0 indexed.

Below I put a commented example of how you can match your different data arrays.

public class Sales{
    public static void main (String []Args) {     
        //a single dimension array containing the following customer names
        String [] Names = {"John Doe","Pete BuysAlot","Joe Stingy","Mary LikesOurStuff" } ;
        //a single dimension array containing the names of each month               
        String[]Months= new String [12]; // <= 12 is the size not the last index !
        Months[0] = "   Jan   "; Months[1] = "   Feb   ";
        Months[2] = "   Mar   "; Months[3] = "   Apr   ";
        Months[4] = "   May   "; Months[5] = "   June  ";
        Months[6] = "   July  "; Months[7] = "   Aug   ";
        Months[8] = "   Sept  "; Months[9] = "   Oct   ";
        Months[10]= "   Nov   "; Months[11]= "   Dec   "; 

         // two dimension array that reflects the purchases each person made per month.
        int[][]slsData = { 
                { 200,   50,   30, 300,  155, 220,  80,  70,  95,  110,   3,   400},
                {1200, 2000, 1500, 900, 1300, 800, 750, 500, 900, 1200, 1500, 2000},
                {  10,    0,    0,  20,    5,  30,  10,   0,   0,   10,   15,    0},
                { 500,  100,  200, 400,  600, 200, 150, 155, 220,  336,  43 ,  455}
        };

        // match the sales data with name & month through the _indexes_ 
        // in the arrays. Here iName [0, 3] and iMonth [0, 11] will
        // give the sales amount for a (name, month): slsData[iName][iMonth]
        for (int iName = 0; iName < Names.length; iName++) {
            System.out.println("\nSales for "+Names[iName]);
            int total = 0; // total (re)set to 0 for each name
            for (int iMonth = 0; iMonth < Months.length; iMonth++) {
                System.out.println(Months[iMonth] + slsData[iName][iMonth]);
                total += slsData[iName][iMonth];
            }
            System.out.println("Total sales for "+Names[iName] + ": "+total);
        }
    }
}