Is there ever a good way to complain about a colleague's negativity?

[Thurlow]

I always feel like mentioning it to management would just be petty or personal.

Just one colleague who moans, flaps, complains and panics constantly about a part of our workload - a part that, ironically, she is used supposed to have some oversight of. It's just constant. And she's complaining about people asking us to do something which is actually part of our role

Sometimes I ignore her, sometimes I put my headphones in, sometimes I try and talk her down but... Is there ever a professional way to say to a manager that a colleague's constant flapping is incredibly hard to work with?

How about "I am concerned that xx isn't managing to deal with [part of job role] very well. She seems to be unsure how to do this part. I think she might benefit from some training/refresh. I don't think she gives a very good impression to [senior management] because she seems so anxious about it. I am worried that she is a bit overwhelmed by things. "

A wise manager will know what you are talking about. Whether they will do anything is another matter.

That's a good idea. Dressing it up as something 'constructive'.