Binary Calculator in Java
Style

I'm sorry for making it personal here, camelCase, but the preferred styling for function names is that they're, well, camelCased. (Apologies for the bad pun). So not GetBinarySum, but getBinarySum.

Removing the Operations class

I think your approach for implementation of Operations is flawed.

You're taking two Strings, converting them to numbers, performing a simple operation and converting it to String again.

This by itself is not a problem, but...

        System.out.println("Sum:        " + operation.GetBinarySum(binOne, binTwo));
        System.out.println("Difference: " + operation.GetBinaryDiff(binOne, binTwo));
        System.out.println("Product:    " + operation.GetBinaryProd(binOne, binTwo));
        System.out.println("Quotient:   " + operation.GetBinaryQuotient(binOne, binTwo));


Here you're doing the following:

read two Strings
convert to ints
sum
convert to binary string
print
convert to ints
diff
convert to binary string
print
convert to ints
prod
convert to binary string
print
convert to ints
quotient
convert to binary string
print


If you instead converted the Strings directly after reading, you could do input validation per input ("hey, you entered a 2 for your second binary number!"), and you'd save on doing all this conversion. You'd convert once, do the math, convert back 4 times (4 different results), and done.

Something like this:

read two Strings
convert to ints
sum
convert to binary string
print
diff
convert to binary string
print
prod
convert to binary string
print
quotient
convert to binary string
print


Here's an implementation without error handling:

        int left = Integer.parseInt(binOne, 2);
        int right = Integer.parseInt(binTwo, 2);

        System.out.println("Sum:        " + Integer.toBinaryString(left + right));
        System.out.println("Difference: " + Integer.toBinaryString(left - right));
        System.out.println("Product:    " + Integer.toBinaryString(left * right));
        System.out.println("Quotient:   " + Integer.toBinaryString(left / right));


And poof! The entire Operations class is gone.

Why have an Operations class

Now, if this were some super advanced calculator which has to do more than just do basic math, it'd help to have the Operations class.

For example, "count all the positive bits in the number" or "detect if this binary number is a square". Then you'd be better served by an operations class that takes int(s) and returns int or boolean. You can later convert this to a String. If you write the implementations like that, your BinaryOperations (which needs a better name) class would be responsible for handling input and output (Like the screens and the buttons of a calculator) and the Operations class would be responsible for doing the actual calculating.

Consider why the Java creators made a java.lang.Math class that takes integers, not Strings and it should be easier to understand how to split responsibilities.

About use of Integer.parseInt with radix

Use built-ins when you can.

Built-ins apply to a larger populace than your custom code. Built-ins have been tested by the people who create the Java language. And above all, built-ins don't require you to implement them yourself.

When the resulting code is written faster, has a greater chance of being correct and uses components people are familiar with, why wouldn't you use built-in functions?

Oh, and in case you forgot what they do, they come with documentation too. So it's easy to remind yourself of how the built-ins work. Custom functions require you to re-read the code if you've forgotten and there's no documentation.