Can I sue my employer 5 months later for a work related injury?

You can potentially sue--you are still within time to sue--but will face two large hurdles in winning:
1) You have to prove that the injury came from this incident and not, say, from something you did on your own time, such as working out or playing sports, lifting furniture or a child or heavy dog at home, slipping and pulling something elsewhere, etc. You will need both credible testimony (e.g. your own) about where the injury occured and medical support for the idea it happened then.
2) You'll have to show that you were not negligent or careless in what you did, because if you were negligent or careless, your own fault will reduce, and possibly eliminate, what you could otherwise recover. You say the gym dd not give you back belts or training in how to safely lift the treadmill; that may be so, but did they stop you from getting and using your own back belt? If you lifted with your back and not legs, for example, how is that their fault, not yours? Etc. You need to also eliminate your own fault as a reasonable cause.
Also, note that you can only recover compensation for medical costs (to date and reasonably foreseeable for the future) and, for significant, long-lasting life impairment or disability (more than "clicking") some amount for pain and suffering. If your doctor visitor shows some significant injury that will impair you and/or reveals the need for extensive physical therapy and/or surgery, then it may be worth considering a lawsuit. But for just a "clicking" sound without evidence of more, it would not be--you would not be able to recover enough compensation to justify the cost of the lawsuit.