If I was exposed to someone with active TB disease, can I give TB to others?

If you also have it. If you inhale the TB bugs and get an active infection yourself, then you can pass it on. If this happens, i'd expect you'd have symptoms like a chronic cough, night sweats, weight loss and maybe a fever.
No. Exposure to someone with active TB does not make you infectious. You can only spread TB to othrs if you yourself actively have tuberculosis. If you were exposed, you should be checked for latent infection.