Celsius and Fahrenheit Converter in JavaScript
Use proper spelling - it'll looks more professional and can prevent bugs stemming from the typo. Farhenhite should be Fahrenheit, and Celcius should be Celsius.
Use proper indentation - every new block should open a new indentation level. You're doing this well with the if/elses, but you should also do it for the whole click listener as well:
converterButton.addEventListener('click', function(event){
    event.preventDefault();

    const celsiusToFahrenheitValue = (celsiusToFahrenheit.value * 9/5 + 32);
    const fahrenheitToCelsiusValue = (fahrenheitToCelsius.value - 32) * 5/9;
    // etc

preventDefault? Is the button inside a &lt;form&gt;? If so, the preventDefault is fine. Otherwise, it won't do anything and can be removed.
Avoid document.write - it can be insecure and somewhat confusing. Here, since it's called after the page loads, it will replace the whole document with the new HTML, which is probably not what you want - you'd probably rather preserve the converter and let people continue to convert after pressing the button once. Instead, populate, say, a results element with the results.
Confusing variable names Without reading the variable definition, it isn't entirely clear what the difference between celsiusToFahrenheitValue and celsiusToFahrenheit.value is. Consider only calculating the new value when it's needed, inside the if statements, and calling it something like convertedToFah. Also consider renaming celsiusToFahrenheit to just plain celsius, since the value it will hold is the plain celcius value.
&lt;h1 class='conversion-type'&gt;&lt;/h1&gt;
&lt;div class='result'&gt;&lt;/div&gt;

if (celsius.value) {
    const convertedToFah = (celsius.value * 9/5 + 32);
    document.querySelector('.conversion-type').textContent = 'Celsius To Fahrenheit Result:';
    document.querySelector('.result').textContent = `${convertedToFah} ℉`;
}

When one input is changed, consider clearing the other input to make it clear what will be converted when the button is pressed - use an input listener.