Will a problem in the liver cause pain in the back?

It can. Yes, this can occur. Also, gallbladder can cause back pain. So can the pancreas and kidney stones. So can muscle spasm, rib dysfunction and a bunch of other things. See your doctor.
Liver & back pain? Generally not. There is too much tissue separating the liver from the spine. The exception is a cancer, which might have a metastasis to both liver and the spine. Similarly, there are other rare infiltrative, metabolic, or disseminated infections that might involve both areas. An exception is a gall bladder problem, which hurts in the same place as the liver; it can send pain to the r shoulder bla.