Something that I forgot to add about Poppy's sudden aggression due to his hormones...If Poppy is in-fact a male (you never know, despite his stereotypical male "hormonal" behavior), and Lilly is in-fact a female (again, same thing), and since Lilly is already over the age of sexual maturity, if you have them housed together in the same cage and they are becoming very closely bonded to one another, this is also fueling Poppy's aggression towards you or anyone else that may come near him, Lilly, or their cage...Think of a hormone-crazed, teenage boy who is trying to protect his girlfriend, but is just acting like an ass...That's basically what Poppy is doing. So besides the fact that you don't want them to start mating/breeding, nor do you want Lilly to start laying eggs continuously, another reason to separate them is to ease Poppy's hormones which will stop his aggression towards you/people...
****Something else that you need to consider, and that every single bird owner who buys first a bird as a pet, and then later-on buys another bird of the same species and of the opposite sex as a second pet:
If your birds are in-fact of the opposite sex (which their behavior does very-much indicate, though you never know for sure without getting them both DNA tested, which you should do ASAP regardless of your issue), yes, it's natural for the male, especially since he's much younger than the female and is just-now going through puberty for the first time, to suddenly go from being a friendly, loving, cuddly little guy to a demon-bird in a split-second, and yes, when he is being housed or is in close, physical contact for most of the day with a female of the same species, this will definitely make his hormonal-aggression a million times worse...And yes, if they are housed together, and even if they are simply allowed to be out of their cages and be physically together a lot of the time, especially unsupervised, then this will eventually result in mating, which will make his aggression much, much worse, as well as result in the female laying eggs, which is VERY BAD FOR HER HEALTH, and unless you have bird-breeding and baby bird hand-raising/hand-feeding experience and education, there's another reason you don't want them to mate...HOWEVER....
****Something that most people are not aware of, nor do they quite understand and certainly don't expect, is that if their 2 birds of the same species and of the opposite sex do in-fact bond-closely with each other and do in-fact start mating with each other and develop a "Breeding-Pair" type of relationship with each other, OFTEN THE RESULT THAT HAPPENS AFTER THE OPPOSITE-SEX PAIR START MATING IS A TOTAL BREAK IN ANY BOND THAT EITHER OF THE BIRDS HAD WITH "THEIR PERSON", OR ANY PEOPLE IN-GENERAL...This isn't something that most people anticipate nor even think about having happen, and once it does they are not only shocked and confused, but it typically also breaks their heart...
Even in the case of Lilly, who you have had as your pet for years, it's very likely that she will lose any bond that she already has with you or anyone in your home if she and Poppy start mating...If they are already close enough that Poppy is showing extreme hormonal aggression towards people trying to touch him, the cage, her, etc., then the next step in this process is typically that the pair start mating...And once that happens, I'd bet at least 90% or more of the time, both birds break any and all bonds that they had with any person or people...This is exactly why "Breeding-Pairs" are advertised for sale as "Breeding-Pairs", and the ads usually say "Not pets"...Even when just a single bird that has been a part of a breeding pair is being sold by itself (usually because it's mate has died), the ad will say something like "Part of a former breeding-pair, not a pet", followed by "Will require someone with the time and patience to work with him to try to tame him again, like he was before he became a breeder"...
Birds are not at all like dogs, cats, rabbits, ferrets, reptiles, etc. They are just like people in a lot of respects, and this is one of them...Where as you can take your male dog to the home of a female dog who is in-heat and that he's never met or seen before, leave the two dogs alone in a room for a while, and typically they will just mate and then separate, and your dog will just get in the car with you and come home, and he'll be the same cuddly, loving dog he was prior to mating with the female dog, BIRDS MUST FIRST FORM A RELATIONSHIP BEFORE EVEN CONSIDERING MATING...We here on this forum are constantly telling people that they can't just put two birds of the same species and the opposite sex together in a cage and expect them to eventually, or ever mate/breed...*****There is a post just like this RIGHT NOW in either the Conure Forum or the Breeding/Hand-Raising Forum that is titled "Why Won't My Green Cheek Conures Breed?", with a totally inexperienced OP who had a female Green Cheek Conure for I believe 3 years, DNA tested female, and who a year and a half ago went to a breeder and bought a male Green Cheek Conure, DNA tested male, who was a member of a prior breeding-pair and was a "proven-breeder" (his mate died I believe, so the breeder was selling him)...This breeder was totally shady and unscrupulous, as many unfortunately are, and they told the OP that this male "would absolutely breed with his female, no problem, it will definitely happen by the end of the year"...Now there is no way at all to know whether any two birds you put together will bond at all...They may absolutely hate each other and must be kept separated at all times for fear of violence, they may not like each other much at all but simply will tolerate each other and are not aggressive, but that's it, they may really like each other a lot but will not closely-bond with each other in the sense that they will mate/breed as they show no hormonal aggression or behavior towards each other or anyone else, they're just "good friends" basically, or they may love each other very much, bond very closely together, and mate/breed...But there was no way for this breeder to know this, they lied to this OP in order to get rid of a single male Green Cheek who's mate died, and who they knew would most-likely never closely-bond or mate with another female and was no use to them...So this OP was asking "What is wrong with my birds? They've been together, living together in the same cage for a year and a half, they like each other and play with each other, but they aren't even close to mating. They have a nest-box, and the male is a proven-breeder...What's wrong?"...Unfortunately we had to tell him that nothing is "wrong", he's not done anything "wrong", he just has a pair of Green Cheeks that are "friends" but that do not want to mate, and at this point if they haven't shown any hormonal behavior towards each other at all, they most likely won't ever...So that OP has the opposite problem that you are having, lol...
I'll continue below and wrap this up very quickly, but since this is a topic that keeps popping-up on here recently over and over again, I'd rather give a long, detailed explanation for people to read if they want to, if not then that's fine too, but at least it's here...It's a topic that a lot of people do not understand and that they aren't at all aware of...