I don't doubt that's where the 45th Space Wing is headquartered, but it appears the Space Command is headquartered out of Colorado Springs but may be moving. From the link you provided it takes me here for Space Command:

https://www.afspc.af.mil/About-Us/Fact-Sheets/

and from that page:

Air Force Space Command Air Force Space Command, activated Sept. 1, 1982, is a major command with headquarters at Peterson Air Force Base, Colorado. AFSPC provides military focused space capabilities with a global perspective to the joint warfighting team.

I don't doubt that's where the 45th Space Wing is headquartered, but it appears the Space Command is headquartered out of Colorado Springs but may be moving.