I do not see "No such file or directory" error. But I see other problems.
Problem #1 - typo
Date today, birthay, aDate;
missing a D from birthday.
Problem #2 - wrong string literal
cout << '\n Chris' birthday : ";
Strings should start with double quotes and end with double quotes. In your code you started with a single quote and ended with a double quote. That not correct.
Problem #3 - missing semicolon
In C/C++ (and many other languages) you must terminate your instructions with a semicolon (;)
time(&sec);
Enhancement
I suggest using the constructor to set up your class, not additional "Init" functions:
class Date
{
private:
unsigned char month;
unsigned char day;
short year;
public:
Date();
Date(unsigned char month, unsigned char day, short year);
void print() const;
};
Furthermore, you can store the month and day on just an unsigned char (1 byte) data structure each.
Last but not least printing the date can be a constant function as there's no need to change the instance while printing it.