#include <iostream>
#include<cstdio>
#include<cstdlib>
#include<Windows.h>
using namespace std;
class Student{
public:
    int age;
    char name[20];
    int Chinese;
    int English;
    int Math;
    Student(int age, char name[20], int Chinese, int English, int Math);
    void WhileGetchar(char name[20]);
    void PrintStudent();
};
Student::Student(int _age, char _name[20], int _Chinese, int _English, int _Math) {
    age = _age;
    name[20] = _name[20];
    Chinese = _Chinese;
    English = _English;
    Math = _Math;
}
void Student::WhileGetchar(char name[20]) {
    int i = -1;
    while (name[i] != '\n') {
        i++;
        name[i] = getchar();
    }
}
void Student::PrintStudent() {
    cout << age << endl << name << endl << Chinese << endl << English << endl << Math << endl;
}

int main()
{
    int age, Math, Chinese, English;
    cin >> age >> Math >> Chinese >> English;
    Student student(age,"**_***",Chinese,English,Math);//报错在这一行
    student.WhileGetchar(student.name);
    student.PrintStudent();
    return 0;
}

因为 "**_***" 是字符串常量，char _name[20] 是可以修改的变量，因此前者不能自动转换为后者。

另外，你的代码错误太多了，
比如 name[20] = _name[20]; 其中 name[20] 和 _name[20] 都不存在
比如 int i = -1; while (name[i] ……，其中 name[-1] 访问了不属于自己的内存。

我帮你改了一下

#include <iostream>
#include <string>

class Student
{
public:
    Student();
    Student( const char* name, unsigned age, double chinese, double english, double math );

private:
    std::string name_;
    unsigned age_;
    double chinese_, english_, math_;

    friend std::istream& operator>>( std::istream& is, Student& st );
    friend std::ostream& operator<<( std::ostream& os, const Student& st );
};

Student::Student()
    : name_(), age_(), chinese_(), english_(), math_()
{
}

Student::Student( const char* name, unsigned age, double chinese, double english, double math )
    : name_(name), age_(age), chinese_(chinese), english_(english), math_(math)
{
}

std::istream& operator>>( std::istream& is, Student& st )
{
    return std::getline(is,st.name_) >> st.age_ >> st.chinese_ >> st.english_ >> st.math_;
}

std::ostream& operator<<( std::ostream& os, const Student& st )
{
    return os << st.name_ << '\n' << st.age_ << '\n' << st.chinese_ << '\n' << st.english_ << '\n' << st.math_;
}

using namespace std;

int main( void )
{
    Student student;
    cin >> student;
    cout << student << endl;
}

输入
输出

谢谢