Palindromes without character
The time complexity of your solution is O(n ^ 2), where n is the length of the given string. It is possible to do much better. Here is a linear solution:

static int getPalindromeIndex(String word) {
    if (isPalindrome(word))
        return -1;
    // Finds the first pair of characters that don't match.
    int left = 0;
    int right = word.length() - 1;
    while (word.charAt(left) == word.charAt(right)) {
        left++;
        right--;
    }
    // One of them must be removed and it sufficient to remove one of them.
    if (isPalindrome(word.substring(0, left) + word.substring(left + 1)))
        return left;
    return right;
}


The idea is the following: 


If it is not a palindrome, there is a pair of positions (i, n - i - 1) such that characters on this positions do not match.
We can remove one of them. It is guaranteed that a solution always exists so we can try to remove the first one and check if a string is a palindrome. If it is the case, then we can just return it. Otherwise we can just return the second one.