Project Euler 25 - x-digit Fibonacci Number
I feel like a class is not a good way to approach this problem. 

C# forces you to write classes, but using static methods will allow you to ignore this.

The only method you should keep non-static is the on calculating the n-th fibonacci, as the memoization requires state, and a class is a way to keep state.

Getting the number of digits of a number is instead something without atate that depends only on the input number, so:

public static digitsLength(BigInteger n) {
    return number == 1 ? 1 : (int) Math.Floor(BigInteger.Log10(number) + 1);
}


Minor remarks are:


Naming: You used get in the name, but in OO get has a precise meaning that is inappropriate here.
Ternary: When the condition is simple, like here, using a ternary increases compactness and readability.


Now main is simply:

for (var i upto infinity) {
    if DigitsLength( Fibonacci(i) ) == REQUIRED_DIGITS {
        print( Fibonacci(i) )
    }
}


PS: Note that the above is pseudo-code, it is just to give you an idea.