Many are familiar with the scenes of astronauts-in-training practicing weightless activities while submerged in a pool with neutral buoyancy.
In this scenario, an astronaut is in such a pool, undergoing such training. This pool is completely contained, essentially a large cube completely filled with water. There is a hatch for egress at the top, usually closed. There is no deck around the pool, no grating or stairs. The water in the cube, however, is not pressurized. It is open to the 'atmosphere' through vents at the top. There are windows around the top, bottom, and sides, for viewing - both from the inside out, and the outside in. Air and communications are continually supplied to her through an umbilical chord. She has a complete astronaut's suit on, fully sealed, pressurized, and self-contained.
So, during one session of her training, this astronaut 'blacks out'.
When she regains consciousness, she looks around and absolutely everything appears to be the same. Same pool, same support swimmers in full gear (she can not see their faces) around her, same environment beyond the pool through the windows that she can see. It appears from her suit chronometer that she was unconscious for only a few seconds.
At what point could she recognize that she was no longer on Earth, but had been transported to an exactly identical facility on the Moon, while unconscious?
There seems to be some confusion about 'neutral buoyancy'. She is the same density as the water. Her pressurized suit, her apparatus, and the density of the water have all been manipulated so that the mass of the water she displaces is exactly equal to her total mass. She neither floats nor sinks. The idea is, that while she is in the water, she is essentially experiencing weightlessness. Any buoyant force pushing her up is exactly countered by the force of gravity pulling her down. When she moves, F=ma is gravity independent. If she moves, the mass of the water she displaces or 'moves aside' is exactly the same mass as the part of her body that is replacing it. Inertia is exactly the same, Moon vs Earth. It requires the same force for her to move in the water on Earth as on the Moon.
The buoyancy force is zero. Be very careful if using formulas and equations, you do not fall into the 'divide by zero' trap. 8-8=0, and so does 5-5=0. If both sides are equal, then no matter how big or small both sides are, the result is still zero. One answer is not bigger or smaller than the other.
Thus, for an acceptable answer, although the tags do not require it, any use of formulas used to 'prove' discrepancies between the situation on Earth and the Moon should include realistic numbers, and a numeric result for Earth and Moon calculations that can be compared. The numbers used have to reflect neutral buoyancy.
Also note, she can not leave the water. There is no 'head room' above the water. She is always 'in' the water.
Note also that she is in a pressurized suit. The suit pressure is automatically adjusted, so it is the same pressure in the water on the Moon as on the Earth.
As provided by Cadence, in the comments, here is a link that might be useful.
SECOND CLARIFYING EDIT
She has no particular reason to suspect that she is no longer on Earth. This is story-dependent, and therefore was not specifically mentioned. She was sedated on the trip. Subtle differences would not make it to her conscious level. It has to be an 'in your face' difference for her to become aware of it. WHY and HOW she has been moved is story-dependent and beyond the scope of the question. However, what is perhaps relevant to the question (in retrospect) is that there is no motivation for her to interpret any subtle differences as being the result of her being on the Moon vs on the Earth. They have to be very noticeable differences that can not be attributed to anything else, in order for her to become aware of them.
THIRD CLARIFYING EDIT
Although this is also plot-dependent, and part of the story line (i.e. not normally relevant), I should clarify that those who brought her to the Moon definitely do not want her to know she is on the Moon. That was the purpose of causing her to 'black out'. However, that is not necessarily relevant to an answer. It is the 'neutrally buoyant' part that is the plot-independent, story-line-independent factor, not anything else that may or may not happen. I am not after an answer that can be 'contrived' by the story line. I am after an answer that can stand independently of any story line.
FOURTH CLARIFYING EDIT
I have found this article as a potential resource. It is a study regarding neutral buoyancy tanks and virtual reality headsets to simulate the 'real thing'. The research was funded/sponsored by NASA.
There are many advantages of training at the NBL. For example, astronauts become accustomed to being confined in the bulky spacesuits. Perhaps more importantly, they experience the sensation of floating as they would in zero-gravity. This is achieved by maintaining neutral buoyancy, meaning that the astronauts do not float to the surface or sink to the bottom. Neutral buoyancy is a good analog for zero-gravity because common sensory cues to body orientation are rendered uninformative. These include somatosensory cues that provide information about pressure on the skin as well as proprioceptive cues that provide information about joint articulation and muscle tension. Both types of cues normally provide information about how the weight of the body is supported. While underwater, only vestibular cues from the inner ear remain to provide reliable non-visual information about the direction of gravity. In this altered sensory environment, astronauts gain valuable experience not only maneuvering in the spacesuits, but also practicing novel locomotion methods.