It can't be proven either way - but Bellatrix wasn’t convicted of murder.
There isn't much information about what Bellatrix did during Voldemort’s first rise to power. She was already a Death Eater and one of Voldemort’s most loyal followers, faithful enough where she didn't denounce him when she was on trial. She certainly could have been one of the Death Eaters who had killed people, and she was undoubtedly capable of being among them, but there's no way of knowing for sure.
However, just because she's capable of killing people, and would be willing to, doesn't mean that she actually did. While being a Death Eater, she would have undoubtedly committed a few crimes, but that doesn't therefore include murder. That she wouldn't be against the idea of killing people doesn't necessarily mean she did. She could have been using other forms of violence. Her not being morally opposed to murder isn't enough evidence that she did kill anyone.
One thing is almost certain - the Ministry didn't have any real evidence that she did kill anyone, otherwise they would have included that in her trial. Her trial was for the torture of the Longbottoms. Murder is a more serious charge, so they wouldn't have left that out of her trial if they had evidence. They might have suspected she did, but without at least some kind of proof that she did, they wouldn't have been able to do anything about it. She certainly wasn’t a convicted murderer. Her conviction was simply over the torture of the Longbottoms - not murder, though some of the Death Eaters were convicted of murder.
“Bellatrix Lestrange, convicted of the torture and permanent incapacitation of Frank and Alice Longbottom.”
- Harry Potter and the Order of the Phoenix, Chapter 25 (The Beetle at Bay)
No one in the Order of the Phoenix ever says anything about Bellatrix having killed people in the First Wizarding War, either. Whenever they talk about the things she did, it's either her torturing the Longbottoms or a general "she was a very evil Death Eater who liked violence and causing pain". But no one in the Order mentioned her killing someone, and they wouldn't need as much evidence as the Ministry to just tell Harry or whoever that they thought she did.
However, that no one had evidence that she killed anyone during Voldemort’s first rise to power doesn't mean she didn't. She could have killed people without them finding out it was her. I doubt that even when people knew something was done by the Death Eaters, that they could trace everything back to the specific Death Eaters who did it.