What is menopause?
Menopause in real-world humans is a biological process in which a female's reproductive system 'shuts down' in an orchestrated way, so that she is no longer capable of bearing children. This is a real puzzle for evolutionary biologists, because the premise of the theory of natural selection is that as organisms evolve they get better at reproducing, so why ever would permanently turning off reproduction be favored? And in fact this intuition seems justified because by some estimates, of 5000 described species of mammals only six are known to have menopause as a life history strategy. Among living primates, only humans are known to have menopause. Currently the most accepted idea for why this might happen is called the "grandmother hypothesis" and it posits that at a certain point in our average woman ancestor's life, she could increase her inclusive reproductive fitness more by caring for her grandchildren than she could by continuing to have her own children. It is interesting that this tradeoff affects biological females so strongly compared to males - while male individual reproductive fitness also peaks and then declines with age, it has a long, gradually decreasing tail rather than a pronounced cliff.
What does this have to do with D&D?
Not much. The game includes some biological processes but completely ignores others. The 'food and water minigame' is prominent - how much food your PC needs, how often, how much that food weighs, what is the kind and quality of food they can receive for different prices, what spells can create or purify food, what happens when they don't get enough food, and so forth all have a place in the rules. On the other hand, there are no rules at all for waste elimination - how many times a day does your PC need to defecate, what that does to travel time if it takes 5 minutes to take off your heavy armor and another 10 minutes to put it back on, how much this waste weighs, whether it is a potential source of disease, and what happens if you don't defecate, are all questions that are just as valid biologically (and are in fact the natural consequence of eating), but they are questions the game chooses not to address.
Menopause is a phenomena associated with aging, and other than rough statements about typical lifespan and age of maturity, the 5e game has no rules about the effects of aging on character traits - even though previous editions did. So even if menopause was actually a thing for female half-elf PCs, we wouldn't have rules for it in 5e.
But would it be a thing? Menopause, as I described above, is thought to be an evolutionary reproductive strategy in a very small number of species of a particular evolutionary history. It is not clear that the 'human' race in the game has had that evolutionary history, or that evolution as we understand it (the change in gene frequencies in a population over time) even exists in the game - or that genes even exist. The fact that humans and elves, two species that appear to have completely different in-game origins, are not only capable of reproducing with one another, but that those offspring are capable of reproduction themselves, means that it is very unlikely that in-game reproductive biology has anything to do with real-world biology. As such, there is no particular reason to think that menopause even exists in-game. There certainly aren't rules for it. So if it does exist in your game, it is only because your DM has decided that it does, and they are responsible for telling you the rules.
The fact that your DM has "refused to comment" means that both you and your fellow players have no reason to believe that menopause is a thing that happens to anyone in your world, let alone to a particular half-elf PC.
A disagreement between the players
I don't understand what is the nature of this disagreement between players. Here are some possibilities, though.
It could be that the other players are simply arguing that "half-elves are menopausal by 37, so the character should be, too," and you are arguing that "no, that is too young, she would have to be older and thus she cannot be menopausal." That's an easy, factual disagreement to resolve - as described above, menopause is not even a thing in your world unless the DM says it is, and they haven't, so any given PC is not menopausal, and likely no one is.
However, it could be that the players in your game feel like they can define each other's characters, and what their character traits are. You might accept that general premise, but disagree with them about this specific assessment of traits.
If you all are doing that in seriousness, then you don't need to justify your opinion on the character's specific traits by an appeal to what you think is a reasonable age for half-elf menopause. There is a larger principle here; this is one person's character. As described in PHB Chapter 1, most of a PC's traits are their player's choice: the race, sex, gender, eye-color, personality, background, and so forth of a character are the player's choice, and not chosen or debated by the other members of the party or even by the DM. This includes whether or not the PC can have children.
Or, it could be that the other players are saying this as a way of making fun of or joking about the character, or teasing the PC's player. Different tables have different playstyles, and such teasing is acceptable at some and not others. Your table has to set its own social norms and to decide whether this kind of behavior is acceptable or inappropriate. If these comments make any of you uncomfortable (especially the player of the half-elf PC, who is on the receiving end), you you don't need to resolve whether or not the PC is actually menopausal; the issue is that some players are engaging in behavior that others don't enjoy. You all need to talk about that.
Is this a double standard?
Although your question does not ask about this, I think it is worth considering. In another question, you ask whether a Paladin's Lay on Hands ability can cure parasites. I found it interesting that in that situation, your fellow PC was trying to help or enable your own PC, while in this question, multiple fellow players seem like they are trying to frustrate one other player. Why the difference?
It might be that in the other question, your (male) PC is engaging in promiscuous sexual activity, and the fellow PC is supporting him in that endeavor. While in this question, a number of your fellow players are saying that a (female) PC is not correctly expressing her reproductive ability, and that it needs to be shut down because she is past a certain age.
We have a long cultural history of being generally supportive of male sexuality, of males having more freedom to express their individual sexuality, but of males being expected to have more sex. At the same time our history encourages more societal control over female sexuality, females have less individual choice about their sexuality, and females are often expected to have less sex. These beliefs, conscious or unconscious, may be part of why you and your fellow players feel like you can debate about or even dictate whether or not a female PC in your game is permitted to have children, while simply accepting that a male PC is successful at being promiscuous just because he wants to be. It might help to resolve your disagreement if you reflected on the internalized nature of this cultural double standard, and you asked your fellow players to reflect on it as well.