Things you should Correct:

1. When the index reaches to the length of inputArray, the inputArray[i+1] will throw an OutOfIndex error, since the indexes' range are from 0 to inputArray.length - 1, and here the last i will have the value of inputArray.length - 1 then the value of i+1 will be inputArray.length and it's not correct.
2. You should make a variable (consider max_res) to save the prevoius maximum result and compare it to new results.
3. Define the prod variable before the for loop, because variables which are first defined in the loop they won't be accessible after the loop ends.
4. Define a vairable to store the index of i that makes the maximum value, since the question wanted the index ,too.

You should change your code like this, to get the correct output:

function adjacentElementsProduct(inputArray) {
  let max_res = Number.NEGATIVE_INFINITY;
  let res_index = -1;
  let prod = 0;
  for (let i = 0; i < inputArray.length - 1; i++) {
    prod = Math.max(inputArray[i] * inputArray[i + 1]);  
    if (prod > max_res){
        max_res = prod;
        let res_index = i;
    }
  }
  return prod;
}

let inputArray = [3, 6, -2, -5, 7, 3]
let prod = adjacentElementsProduct(inputArray)

console.log(prod);

1) You have to get the max of two numbers as

sum = Math.max(sum, inputArray[i] * (inputArray[i + 1] ?? 1));

2) You also have to handle the last case where i + 1 will be undefined then you have multiply by 1. You can use null coalescing operator

(inputArray[i + 1] ?? 1)

function adjacentElementsProduct(inputArray) {
  let sum = 0;
  for (let i = 0; i < inputArray.length; i++) {
    sum = Math.max(sum, inputArray[i] * (inputArray[i + 1] ?? 1));
  }
  return sum;
}

const inputArray = [3, 6, -2, -5, 7, 3];
console.log(adjacentElementsProduct(inputArray));

Number.MIN_SAFE_INTEGER will allow the smallest integer because this needs to find the largest sum which may include negative integers.

function solution(inputArray) {
    let largestProduct = Number.MIN_SAFE_INTEGER;
    for (let i = 0; i < inputArray.length; i++) {
        if(inputArray[i] * inputArray[i + 1] > largestProduct) {
            largestProduct = inputArray[i] * inputArray[i + 1];
        }
    }
    return largestProduct;
}