1) Why do you have class here? What is the purpose of timcalcobject? It does not contain any data. Why not make it standalone functions in namespace/static functions of class?
2) You have repeating code. For example getting current date and getting birthday date.
Move it in own functions and call when needed.
3) conio.h is not needed here (and my advice is to avoid it), numday variable is not used.
4) If a function body does not fit on screen, then the function is too long. Consider separating it into logical blocks.
5) Use arrays for logically bound data. For example number of day in month can be represented by integer array. And amont of days passed can be calculated from current month index by loop.
http://pastebin.com/fvV9KjRj
Not the most elegant solution, and some of the fucntions are actually do not belong here (I copypasted them from my "console.h"), but it works, and has some cool new functions.
It requires C++14. If your compiler does not support it, delete line usingnamespace std::literals; and replace all ""s with std::string("")