Why isn't Montgomery modular exponentiation considered for use in quantum factoring?
Could you post the original Japanese title/reference?
Also, you might consider just writing to the author - assuming it's the same guy he is a professor at the University of Tokyo:
http://www.it.k.u-tokyo.ac.jp/~kunihiro/
and almost certainly would reply.
Sorry to post this as an answer, it should be a comment but I don't have the rep for that yet apparently...
EDIT:  So, I took a look at the original Japanese.  As a preface, I am currently a PhD student in the EE department at U. Tokyo, originally from the U.S., and I do technical JA-&gt;EN translation as a part-time job. However, this topic area is well outside my comfort zone so please take my opinion with a grain of salt!
Basically the conclusion (4) says:

べき乗剰余演算を行う量子回路を提案した。提案方式は、右向きbinary methodを採用し、Montgomery Reductionを採用するという特徴を持っている。従来の方式よりも、回路の構成要素が少ないという特徴を持っている。qubitが多く必要となるという欠点は持つが、より少ない計算時間で計算ができると期待される。
[In this paper] We proposed a new quantum circuit for computing modular exponentiation. The proposed method utilizes an LR binary method, and is also characterized by use of the Montgomery Reduction.  Compared with previous methods, the proposed method requires fewer components to construct the circuit.  The proposed method does however, have the drawback of requiring a large number of qubits, but we are confident that it will be computationally efficient ( lit: require very little computation time ).

I tried searching for related follow-up papers in both English and Japanese but was unsuccessful.   My guess is that the approach proved unsuccessful, or the professor got busy with something else (looks like this was around when he switched universities).
I think that your best bet at this point, assuming you want to follow up the rest of the way and get a concrete answer, is to write professor Kunihiro directly (in English!)