Uncaught TypeError: Function is not a function

The real problem is that the this inside event handlers is not what you think it is. The this inside the event handler will be the (DOM) element that fired the event, not the instance of your class.

Now while you were trying to fix the real problem, you ended up with another problem, which is that you are shadowing the class methods with properties that have as values the empty string ''.

To fix this, just remove the constructor all together as it doesn't do anything, and fix the issue with the this inside the event listener. To do that you have plenty of ways:

1. Use a variable outside the event listener scope called, for example, that, assign this to it and use that instead of this inside the event listener.

Like so:

var that = this;
field.addEventListener("keyup", function(e) {
    // use 'that' instead of 'this'
    if(that.isInt(valueParsed) ...
});

2. Use an arrow function because arrow functions use their surrounding this value:

Like so:

// notice the arrow function passed to addEventListener
field.addEventListener("keyup", e => {
    // you can now use 'this' here with no problems
    if(this.isInt(valueParsed) ...
});

1. bind your event handler to the instance of your class. binding a function will create a new function that always have its this value set to whatever you set it to.

Like so:

field.addEventListener("keyup", function(e) {
    // you can now use 'this' here with no problems
    if(this.isInt(valueParsed) ...
}.bind(this)); // bind the function to its surronding 'this' value so 'this' inside it will be the same as 'this' outside it

Working code: Using an arrow function

class Fibonacci {
  inputValidate(valueParsed, isInt) {
    var field = document.getElementById('fibonacci');
    var valueParsed = parseInt(field.value);
    field.addEventListener("keyup", e => {
      if (this.isInt(valueParsed) === false && field.value !== '') {
        alert('Please enter a valid integer.');
      }

      if (this.isFibonacci(valueParsed)) {
        alert(valueParsed + ' is a Fibonacci Number.');
      } else {
        alert(valueParsed + ' is not a Fibonacci Number.');
      }
    });
  }

  isInt(valueParsed) {
    var field = document.getElementById('fibonacci');
    var valueParsed = parseInt(field.value);
    return !isNaN(valueParsed) && valueParsed == valueParsed.toFixed();
  }

  isPerfectSquare(valueParsed) {
    var field = document.getElementById('fibonacci');
    var valueParsed = parseInt(field.value);
    var squaredValue = parseInt(Math.sqrt(valueParsed).toFixed(0));
    if (field.value !== '') {
      return (squaredValue * squaredValue == valueParsed);
    }
  }

  isFibonacci(valueParsed) {
    var field = document.getElementById('fibonacci');
    var valueParsed = parseInt(field.value);
    var squaredValue = parseInt(Math.sqrt(valueParsed).toFixed(0));
    return this.isPerfectSquare(5 * valueParsed * valueParsed + 4) || this.isPerfectSquare(5 * valueParsed * valueParsed - 4);
  }
} // class

let Fib = new Fibonacci();

<form id="fibonacci-form" action="" method="post">
  <input id="fibonacci" type="text" name="fibonacci" />
</form>

Enhanced working code:

There still are some issues with your code that are related to the functionality rather than errors:

1. You declare functions as having arguments but you are not using them. For instance, the valueParsed argument is not used at all by even though it is an argument in all functions, instead you are fetching it from the DOM element every time. Use the argument.
2. The valueParsed (now used as argument) will be initialized inside inputValidate. It should be fetched from within the event listener not from outside it (every time the event fires we should get a new value for valueParsed).
3. For validation use Number instead of parseInt if you want to exclude float numbers (using parseInt will let float numbers pass validation as it takes only the integer bit from them). Also if validation fail, return to stop further code from executing. It (validation) still not very good though, I'll leave that to you.
4. Suggestion: You may want to use a button and listen for clicks on it instead of listening for keydown input on the field which is annoying. Create a button, and when the user clicks the button then check if the number they entered in the field is a Fibbonacci one or not. You would only change a line or two of code to achieve that.

class Fibonacci {
  inputValidate() {
    var field = document.getElementById('fibonacci');

    field.addEventListener("keyup", e => {
      var valueParsed = Number(field.value);
      if (this.isInt(valueParsed) === false) {
        alert('Please enter a valid integer.');
        return;
      }

      if (this.isFibonacci(valueParsed)) {
        alert(valueParsed + ' is a Fibonacci Number.');
      } else {
        alert(valueParsed + ' is not a Fibonacci Number.');
      }
    });
  }

  isInt(valueParsed) {
    return !isNaN(valueParsed) && valueParsed == valueParsed.toFixed();
  }

  isPerfectSquare(valueParsed) {
    var squaredValue = parseInt(Math.sqrt(valueParsed).toFixed(0));

    return (squaredValue * squaredValue == valueParsed);
  }

  isFibonacci(valueParsed) {
    var squaredValue = parseInt(Math.sqrt(valueParsed).toFixed(0));
    return this.isPerfectSquare(5 * valueParsed * valueParsed + 4) || this.isPerfectSquare(5 * valueParsed * valueParsed - 4);
  }
} // class

let Fib = new Fibonacci();

<form id="fibonacci-form" action="" method="post">
  <input id="fibonacci" type="text" name="fibonacci" />
</form>