Only if the DM decides it does
There is no explicit rule in the game which states that becoming prone would make someone fall if they were currently climbing, and it's not an automatic direct consequence of any of the text in the prone condition itself. It'd be a consistent interpretation of the rules that being prone, whilst climbing, reflects a state of being off-balance or barely holding on but not quite falling.
Contrast this with the rules for flying movement, which do explicitly state:
If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic, such as by the fly spell.
The fact that this is stated with regards to flying movement but no equivalent rule exists for climbing lends support to the reading that being prone doesn't automatically cause a climber to fall.
This also appears to be the reading arrived at by the designers. As we all know, Jeremy Crawford's tweets aren't official rules clarification anymore, but he was asked about this in 2017 and responded:
No general rule causes a climbing creature to fall if knocked prone. As DM, I'd look at the environment and decide. #DnD
So the designer interpretation here is that it's not an automatic consequence of the rules, but it's within the DM's purview to judge that in any particular situation, becoming prone might have the additional consequence of a dangerous fall.
In my personal judgement that might vary depending on the creature in question, and how exactly the prone condition was inflicted. Becoming prone because of hideous laughter is a much gentler experience than becoming prone because Bob the Battlemaster hit you in the face with a hammer and a trip attack, for instance.