Perform a SSH connection with Windows API in C/C++?

I have not used this myself but libssh looks like it would fit the bill for you.

There is no native SSH client (or library) in Windows. If you're not looking to do it from scratch then I'd look again at PuTTY and it's associated programs. these can be distributed with your application so won't require an extra installation. It's also open source so (dependant on licencing) you may be able to reuse/port some of the code)