Yes, although sometimes east Washington is also called inland Northwest but is distinctly not part of the mountain west. A lot of the "who is what Northwest" confusion comes from which group a person wants to associate with or other from more. People in Seattle typically call anything east of the Cascades inland Northwest, people in the Palouse area often refer to themselves as Pacific Northwest so naturally that includes all of Washington, Idaho is in a perpetual identity crisis because of the large cultural difference between North and South Idaho. People in South Idaho refer to themselves either as inland Northwest to be more associated with Washington/Oregon (but not be so left leaning as to associate with the pnw) or as Midwest if they're culturally closer to the farmers and truckers coming out of the actual Midwest.

I'm used to people saying Rockies when talking about the mountains themselves. See other comments about calling the region the Mountain West though, or Mountain States, which are named that because of the Rockies