What you actually want to test is
if (number % 3 == 0 && number % 5 == 0) …
else if (number % 3 == 0 && number % 5 != 0) …
else if (number % 3 != 0 && number % 5 == 0) …
else if (number % 3 != 0 && number % 5 != 0) …
if you'd write out the four cases.
Only you don't need to be that explicit, because when the previous conditions already did not match (and you are in the else branch), then those != 0 are implied and you can omit them. However, order matters, as the conditions are tested consecutively.
So if you have the fully qualified conditions, you can shuffle their order as you want:
if (number % 3 == 0 && number % 5 != 0) … // move to front
else if (number % 3 != 0 && number % 5 == 0) …
else if (number % 3 == 0 && number % 5 == 0) …
else if (number % 3 != 0 && number % 5 != 0) …
and then continue to simplify conditions, omitting the parts that are already implied by their parent cases:
if (number % 3 == 0 && number % 5 != 0)
console.log("Fizz");
else if (number % 3 != 0 && number % 5 == 0) // (an == instead of the && would suffice)
console.log("Buzz");
else if (number % 3 == 0) // as it didn't match the first condition, we know that % 5 == 0
console.log("FizzBuzz");
else // here we know that % 3 != 0 && % 5 != 0
console.log(numer);
Other permutations of the condition let us use as few as in your original example, like
if (number % 3 == 0 && number % 5 != 0)
console.log("Fizz");
else if (number % 3 == 0) // as it didn't match the first condition, we know that % 5 == 0
console.log("FizzBuzz");
else if (number % 5 == 0) // as it didn't match the first condition, we know that % 3 != 0
console.log("Buzz");
else // here we know that % 3 != 0 && % 5 != 0
console.log(numer);
And the minimum number of tests would be achievable by nesting them:
if (number % 3 == 0)
if (number % 5 == 0)
console.log("FizzBuzz");
else
console.log("Fizz");
else
if (number % 5 == 0)
console.log("Buzz");
else
console.log(numer);