If there was a violent crime, it'll be listed with the police. So you can ask the police department whether any activity has occurred there.
But if someone died of a heart attack, or stroke, or old age . . . no, there wouldn't be any way to determine that.
If you strongly suspect that there's been a death, you could also ask the neighbors. They might know. But if you're ashamed to ask the landlord, I suspect you may not be up to asking the neighbors.
And where do you draw the line? Someone has a heart attack. The paramedics come. The patient is unresponsive, but is taken to the hospital where he's pronounced dead.
Or someone was dying of cancer. A few days before her death, she was taken to the hospital and died there.
Unfortunately, people die. They all die. It's something you have to deal with.