Can I sue a store for selling me a mattress that caused a back pain?

To win a lawsuit, you would need evidence that the mattress caused you back pain. That's more than just you saying you never had back pain before--the court will not and cannot assume that the mattress is the cause, when you could have slipped, tripped, pulled something exercising or lifting something, could be developing arthritis, etc. So you would need a doctor to testify that he or she examined you and that in his or her expert opinion, the mattress is the cause.
Furthermore, you'd also need evidence that the mattress is defective in some way: e.g. designed or manufactured improperly. That would require someone knowledgeable or with relevant experience or eduction examining the mattress and explaining in court how it is defective. After all, if the mattress is not defective, but you bought the wrong mattress for you, that is not the store's fault and they are not liable for it. (For example: I develop back pains if on too soft a mattress, even if it is perfetly good one; I need firmer mattresses with more support, and if I bought a soft mattress and experienced pain, that is not the store's fault or issue.)
So the answer is, yes you can sue, but you have to be able to prove that a defect caused your back pain (and that nothing else caused your pain), and that will require one or two expert witnesses (e.g. a doctor). It is an expensive undertaking--you cannot get your expert witness cost back, but must pay it yourself--even if you do not hire an attorney. And winning is not guaranteed (no case is)--it depends on the strength of your evidence. You may be better off selling the mattress on eBay or Craigslist, etc. to recoup some of your cost.