Yes bed bugs can and do travel from room to room on their own even when you didn't transfer them.
Bed Bugs can crawl from one infected area to another area and infect other rooms so if you have bed bugs in one room it's a good idea to exterminate other rooms for bed bugs as well especially rooms that are next to the infected room.
Bed Bugs can be spread through being on peoples clothing however the most common way that bed bugs spread from place to place are in luggage, backpacks, briefcases, mattresses, and used furniture.