直接上代码（注意这里使用了Hutool里面的DateUtil工具类）：

import cn.hutool.core.date.DateTime;
import cn.hutool.core.date.DateUtil;

import java.text.ParseException;
import java.util.Date;

public class DateUtilPlus {


   public static Integer getWeek(String beginDateStr, String endDateStr) {
        DateTime beginDate = DateUtil.parseDate(beginDateStr);
        DateTime endDate = DateUtil.parseDate(endDateStr);
        DateTime currentDate = DateUtil.dateNew(new Date());
        int week = 1;
        if (DateUtil.isIn(currentDate, beginDate, endDate)) {
            Date endDateOfWeek = DateUtil.endOfWeek(beginDate);
            while (!DateUtil.isIn(currentDate, beginDate, endDateOfWeek)) {
                ++week;
                endDateOfWeek = DateUtil.endOfWeek(DateUtil.offsetDay(endDateOfWeek, 1));
            }
            return week;
        }
        return null;
    }

    public static Integer getMonth(String beginDateStr, String endDateStr) {
        DateTime beginDate = DateUtil.parseDate(beginDateStr);
        DateTime endDate = DateUtil.parseDate(endDateStr);
        DateTime currentDate = DateUtil.dateNew(new Date());
        int month = 1;
        if (DateUtil.isIn(currentDate, beginDate, endDate)) {
            Date endDateOfMonth = DateUtil.endOfMonth(beginDate);
            while (!DateUtil.isIn(currentDate, beginDate, endDateOfMonth)) {
                ++month;
                endDateOfMonth = DateUtil.endOfMonth(DateUtil.offsetDay(endDateOfMonth, 1));
            }
            return month;
        }
        return null;
    }

    public static void main(String[] args) throws ParseException {
        Integer week = DateUtilPlus.getWeek("2020-08-01", "2020-10-31");
        Integer month = DateUtilPlus.getMonth("2020-08-01", "2020-10-31");
        System.out.println("week--->" + week + "  month--->" + month);
    }