The previous answers looks ok, but lacks one major thing when dealing with array indexes ; validation checks.
You do not want to try to access non-existent array indexes. Therefore, I created a small class to shift your array as needed, with validation. This will throw an Error if either the row or column index is invalid.
class ArrayShifter {
    static showArray(array) {
        // console.log("Array : ", array);
        console.log('------');
        for (const [index, elem] of array.entries()) {
            console.log(''+elem);
        }
    }
    static validateRowIndex(array, rowIndex) {
        if (!isArray(array) || !isInt(rowIndex) || rowIndex <= 0 || rowIndex > array.length) {
            throw new Error('The row index is wrong');
        }
    }
    static validateColumnIndex(array, columnIndex) {
        if (!isArray(array) || !isInt(columnIndex) || columnIndex <= 0 || columnIndex > array[0].length) {
            throw new Error('The column index is wrong');
        }
    }
    static shiftRow(array, rowIndex) {
        ArrayShifter.validateRowIndex(array, rowIndex);
        array[rowIndex - 1].unshift(array[rowIndex - 1].pop());
        return array;
    }
    static shiftColumn(array, columnIndex) {
        ArrayShifter.validateColumnIndex(array, columnIndex);
        let prev = array[array.length - 1][columnIndex - 1];
        for (const elem of array) {
            let tmp = elem[columnIndex - 1];
            elem[columnIndex - 1] = prev;
            prev = tmp;
        }
        return array;
    }
}
let sourceArray1 = [
    [1,2,3],
    [4,5,6],
    [7,8,9],
];
let sourceArray2 = [
    [1,2,3],
    [4,5,6],
    [7,8,9],
];
let controlArrayShiftRow = [
    [3,1,2],
    [4,5,6],
    [7,8,9],
];
let controlArrayColumnRow = [
    [7,2,3],
    [1,5,6],
    [4,8,9],
];
// arrayShifter.showArray(sourceArray1);
console.log(`Shift row test is ${areArraysEqual(controlArrayShiftRow, ArrayShifter.shiftRow(sourceArray1, 1))}.`);
// arrayShifter.showArray(sourceArray2);
console.log(`Shift column test is ${areArraysEqual(controlArrayColumnRow, ArrayShifter.shiftColumn(sourceArray2, 1))}.`);
//-------------------- Unimportant js functions --------------------
function isArray(arr) {
    if (Object.prototype.toString.call([]) === '[object Array]') { //Make sure an array has a class attribute of [object Array]
        //Test passed, now check if is an Array
        return Array.isArray(arr) || (typeof arr === 'object' && Object.prototype.toString.call(arr) === '[object Array]');
    }
    else {
        throw new Exception('toString message changed for Object Array'); //Make sure the 'toString' output won't change in the futur (cf. http://stackoverflow.com/a/8365215)
    }
}
function isInt(n) {
    return typeof n === 'number' && parseFloat(n) === parseInt(n, 10) && !isNaN(n);
}
function areArraysEqual(a1, a2) {
    return JSON.stringify(a1) == JSON.stringify(a2);
}
The working code can be seen in this codepen.