Police were called just after 10:15 GMT on 14 January after reports of an "altercation" between the workers.
No injuries were reported and no formal complaint was received from either party, a spokesman said.
Betsi Cadwaladr University Health Board (BCUHB) said it was carrying out an internal investigation.
Carol Marubbi, president of Llandudno Hospital Action Group, said the alleged incident had taken place on the Morfa ward, which treats people recovering from strokes.
"I was appalled when I heard about this," she said.  "You don't expect this sort of behaviour in a hospital, especially on a rehabilitation ward.
"The people there are so vulnerable. They've had strokes and brain injuries and to witness something like that, it's an utter disgrace."
A North Wales Police spokesman said: "We were called to reports that there had been an altercation between two members of staff at Llandudno Hospital earlier in the morning.
"This matter is now being dealt with by the hospital authorities and not North Wales Police."
A spokesman for BCUHB said: "We are aware of an alleged incident and are carrying out an internal investigation."
An investigation is under way after two staff were involved in an alleged fracas on a ward in Llandudno Hospital, Conwy.