Can we take the Trans-Siberian Railway in HKSAR?

Yes, but it is not direct through train ineed.
First, you have to on board T98 train which is the service between Hong Kong to Beijing West. Once you get to the capital city, take K3 train depart from Beijing to Moscow via Siberia.

圖片參考：http://blog.donews.com/images/blog_donews_com/prclixiao/101895/r_134812981_6a54590680.jpg

you need to change in beijing.

Yes. But you need to take the train to Beijing first.