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Use TryParse() for retrieving integer input from the console. We don't want exceptions on user input.

You can change your GetGCD() method to be much more intuitive and minimal using a ternary operator and recursion. I recommend also documenting what it's doing for readability purposes:

static int GetGCD(int x, int y)
{
    // if y is equal to 0, return x.
    // if y is not equal to 0, recursive call with x as y and y as the remainder.
    return y == 0 ? x : GetGCD(y, x % y);
}

Modular arithmetic should be positive. To ensure this, we use the absolute value of the input (Euclidean division). Here's how I would implement it:

static void Main()
{
    int x, y;
    
    // Get x. Ensure it's an integer.
    Console.WriteLine("Input first number: ");
    while(!Int32.TryParse(Console.ReadLine(), out x));
    
    // Get y. Ensure it's an integer.
    Console.WriteLine("Input second number: ");
    while(!Int32.TryParse(Console.ReadLine(), out y));

    // Ensure the values being used are positive.
    Console.WriteLine(GetGCD(Math.Abs(x), Math.Abs(y)));

    Console.ReadKey();
}

The resulting console output:

Input first number: 
a
b
c
585
Input second number: 
d
442
13