Quantum polynomial hierarchy vs counting hierarchy First of all, I'm kinda surprised that I couldn't find any paper/article defining such hierarchy.

It can be defined as follows:

$\Delta_0^{\mathsf{BQP}}=\Sigma_0^{\mathsf{BQP}}=\Pi_0^{\mathsf{BQP}}=\Delta_1^{\mathsf{BQP}}\mathsf{=BQP}$

$\Delta_k^{\mathsf{BQP}}=\mathsf{BQP}^{\Sigma_{k-1}^{\mathsf{BQP}}}$

$\Sigma_k^{\mathsf{BQP}}=\mathsf{QMA}^{\Sigma_{k-1}^{\mathsf{BQP}}}$

$\Pi_k^{\mathsf{BQP}}=\mathsf{coQMA}^{\Sigma_{k-1}^{\mathsf{BQP}}}$

Is this whole hierarchy known to be contained in some finite level of counting hierarchy like $\mathsf{PH}$ (which is contained in the second level of counting hierarchy, $\mathsf{P^{PP}}$)? If not, are there some expectations on it?
I was quite surprised as well to not find this hierarchy in the literature, so I wrote my graduate thesis about it. It will be available online soon, at which point I will update this answer with a link to it. I was able to prove some interesting theorems about it.

Theorem For every $i$, $\Sigma_i^{\mathsf{BQP}}\subseteq C_i^\mathsf{P}$, where $C_i^\mathsf{P}$ is the $i$-th level of the counting hierarchy. So the whole hierarchy is contained in the counting hierarchy.

Proof Trivially from $QMA\subseteq PP$. $\square$

Theorem If $\Delta_i^{BQP}=\Sigma_i^{BQP}$ then $BQPH=\Delta_i^{BQP}$.

Proof Excercise for the reader.

Theorem If $QMA=PP$ then $PH\subseteq QMA$ and indeed $PH^{PP}=QMA$.

The first part of this theorem was originally by Vyalyi [4], but his proof is six pages of algebra, whereas new ideas that originated from thinking about your hierarchy allow me to get there in just a few lines:

Proof I prove a new theorem: $QMA\cap coQMA=P^{QMA\cap coQMA}$. This gives us $QMA=P^{QMA}=P^{PP}\supseteq PH$ by Toda. $\square$

For the new part, $PH^{PP}=QMA$, you will have to read my thesis. There's no obvious analogue of Toda's theorem. The reason is that the critical ingredient $QMA\subseteq BPP^{\oplus P}$ seems hard in the quantum case, whereas classically $NP\subseteq BPP^{\oplus P}$ follows trivially from Valiant-Vazirani, because here we don't have a way to "count" solutions. Even if you take $QCMA$ instead of $QMA$, there's the inconvenience that a QCMA protocol can behave arbitrarily on bad certificates to yes-instances. To see what I mean, start with [1].

There's also no obvious analogue of a Karp-Lipton collapse, or Kannan-type circuit lower bounds. Recall that in the classical case, we have

Theorem [3] For every $k\in\mathbb{N}$, there is a language in $\Sigma_4^{P}$ that requires circuits of size $\Omega(n^k)$.

The reason there's no(t yet) a theorem replacing every $P$ in this theorem with $BQP$ is that a critical ingredient in his proof is that he is able to (non-deterministically) guess a circuit and then evaluate that circuit, i.e. compute the circuit's output on an input. Sure, we can guess a (description of a) quantum circuit, but we can only say that we evaluate it if it is a bounded-error circuit, and verifying this is a $\# P$-Hard problem.

Scott Aaronson and Andy Drucker prove something along the lines of a "quantum PH", going only to the second level:

Theorem [2] If $NP\subset BQP_{/qpoly}$ then $\Pi_2^P\subseteq QMA^{PromiseQMA}$.

When I asked Scott Aaronson, he said that people had been thinking about this hierarchy for 15+ years, but nobody had published it because nobody had proved anything non-trivial about it. Here's to hoping you or I change that.

Bonus Theorem! If $QMA=coQMA$ then $PH\subseteq QMA$. Just for you, because you made it all the way to the end of a long answer.

[1] Aharonov, Dorit, et al. "The pursuit for uniqueness: Extending Valiant-Vazirani theorem to the probabilistic and quantum settings." arXiv preprint arXiv:0810.4840 (2008).

[2] Aaronson, Scott, and Andrew Drucker. "A full characterization of quantum advice." Proceedings of the forty-second ACM symposium on Theory of computing. ACM, 2010.

[3] Kannan, Ravi. "Circuit-size lower bounds and non-reducibility to sparse sets." Information and Control 55.1-3 (1982): 40-56.

[4] Vyalyi, Mikhail N. "Qma= pp implies that pp contains ph." In ECCCTR: Electronic Colloquium on Computational Complexity, technical reports. 2003.