The class number of integral binary quadratic forms of discriminant $D$ correspond to the class number of $\mathbb Q(\sqrt{D})$. In this case the class number formula is the classical one of Dirichlet.

For the class number of binary hermitian forms of discriminant $D$, on the other hand, there is no such correspondence I am aware of. Still, there is a similar formula proved by G. Humbert around 1920, but it does not give $h(D)$ itself (see here). In a special case: $$ \sum_{j=1}^{h(D)}\frac{D}{4}\sigma(F_j)=\frac{\pi}{4}\prod_{p|D,p\neq2}(1+\Big(\frac{-1}{p}\Big)\frac{1}{p}) $$ where $\sigma(F_j)$ is a certain collection of fundamental domains.

I'd like to know if there is a more precise formula available.