I'm fairly sure I've read in several places that traumatic hospital care as an infant can continue to reverberate. Traumatic experiences as an infant can continue to affect the child as they grow up.
It's about trauma (in terms of how PTSD works) rather than "remembering" as such. I seem to recall an interview on the BBC website with a woman diagnosed with PTSD as an adult from a hospital admission that occurred when she was a baby and that she had no conscious memory of.
So, whilst you don't remember the experience, your brain remembers that you previously felt threatened/unsafe/pain the last time you encountered a hospital bed or the smell of antiseptic etc and then reacts as if you are in danger every time you encounter those things.
It takes longer for your brain to relearn that those things can be encountered safely - it takes a prudent approach to protecting you from danger.
I won't pretend to know how you can help a one year old to heal from something like this. I'm not even sure where you could seek professional advice in terms at least of not inadvertently making it worse. My comments are based on adults with trauma (whether the trauma happened as an adult or child).
Making sure she feels safe, comforted and protected whenever she faces a stimulus that frightens her like this will be important. I doubt trying to force her to accept it whilst frightened will help (more likely just to reinforce the threat association in her brain), but having you make her feel safe should help.
Eg, don't make a big deal of the bed, try treating it as something you're not interested in or bothered by while you just happen to be in the room with it doing something calm, safe, soothing and nice together (could you take her into the room to do something that doesn't involve approaching the bed? Like playing in a corner or something?). That way she learns she can be around those beds without anything nasty happening to her. The more experiences of safety the better to help her brain relearn it's not a threat.