Well, a valid C++ program (or really any C++ program) will simply be a finite sequence composed of a finite collection of characters and a few other things (indentation, spaces, etc.). It is a general result that the set of all finite sequences of entries from a finite alphabet will be countably infinite. To show that there are countably infinitely many valid C++ programs, you need only show there is no finite upper bound on the length of valid C++ programs.
Addendum: Another approach (an alternative to showing there is no finite upper bound on length) is to actually explicitly define (in a theoretic sense) countably infinitely many valid C++ programs. For example, for a given positive integer, the program that simply prints said integer, then ends (as I mentioned in the comments below).
The following program template should do the trick:
using namespace std;
int main ()
cout << "___________";
That "____" part is the spot where you'd type in whatever positive integer you wanted the program to print out--whether that be $1$, or $23234$, or $1763598730987307865$, or whatever--instead of the underscores.
Now, obviously, no matter how fast you can type, there are integers big enough that you couldn't finish typing in your lifetime, so in practice, there are programs of this type that you could never finish. Even if such a program were handed to you, you'll certainly run into memory problems for sufficiently large integers (depending on the computer), but should still be valid programs. We can say that such programs all exist in a "theoretical" sense. That is, given sufficient memory and power to store and run it--necessarily a finite (though perhaps prohibitively large) amount--and given sufficient time to program and run it--necessarily a finite (though perhaps prohibitively long) amount--this program will do what it's supposed to do.
Please don't give me any grief about the heat death of the universe or anything like that. ;)