Q:

Why ostream is not convertible to ostream?

I am doing some type checking in my project. The following example 
using namespace std;

cout << ( is_convertible<ostream,ostream>::value ? "TRUE":"FALSE" ) << endl;

returns "FALSE".
Could someone explain why?

A:

is_convertible tests if an imaginary function
 To test() { return std::declval<From>(); }

is well formed (source).
std::ostream is an alias for a std::basic_ostream<char>.  Its copy constructor is deleted, and its move constructor is protected.
So the test imaginary function is not well formed.
In short, you asked if you can move construct an ostream from an ostream, and the answer is "no".

A:

You can't copy an ostream (it would make no sense in terms of internal FILE* representation). So you want to test with references instead (which is always a little safer than pointers):
std::cout << std::is_convertible<std::ostream&,std::ostream&>::value;

