Yes, counseling can work in many situations without delving into a client's childhood. When we hear the words "counseling" or "therapy" an image of a comfy couch with extensive psychoanalysis comes to mind for many. In current times, this type of therapy is rare.
Many counseling techniques utilized by mental health practitioners now are goal-oriented and structured. If delving into the client's childhood will not have a positive impact on the client's current ability to develop more effective adaptive strategies and cope with daily stressors successfully, it is not necessary.
I do want to note that if a client is seeking counseling yet is very resistant and anxious to discussing their childhood, it is unfortunately likely that something in the client's childhood needs to be resolved in order for the client to deal with the present issue effectively.
Can counseling be effective without talking about your childhood? Sure, but it is also possible for many people that avoiding their childhood, if this is where the root of the problem lies, could set up the person for continued failed counseling attempts.