I've done numerous nebulizer treatments on my birds for upper respiratory infections, I even built a custom treatment tank that the bird/beardie goes in out of a plastic tub that the nebulizer feeds into. In my personal opinion, if a beardie, bird, etc. has an upper respiratory infection, the most efficient and effective way to deliver antibiotics to them is by nebulizer. It's far more effective than oral antibiotics, and every time I've given them a nebulizer treatment using antibiotics they seem to really feel much better within 24 hours. So it's wonderful that the vet suggested this treatment to you. Using F10 in a nebulizer is a fairly new way of treating respiratory infections, but so far has shown to be very effective.
As far as preparing the antibiotics, there are a few different ways you can make up the solutions for the nebulizer. Sometimes the vet or pharmacist will actually dilute the already liquid antibiotic to the correct concentration and you can just put the antibiotics directly into the nebulizer, sometimes the vet or pharmacist will prepare a solution that is ready to use out of powder antibiotics or they will actually make powder out of antibiotic pills and then prepare the solution. In all of these cases you need not dilute, you just put the pre-prepared solution into the nebulizer, so you always want to make sure that you have to dilute what you've been given and that it it wasn't already done. If you need to dilute the liquid antibiotics you have, you can use any number of things to do so. I've used distilled water, sterile saline, sterile water, and even carrier oils in some situations, depending on what antibiotic you're using. For Gentomycin you can use either distilled water, sterile water, or sterile saline, any of these will work fine and be safe for your beardie to breathe in. You can buy all 3 at any drugstore or even at Walmart, I believe the last bottles of sterile water and sterile saline I bought were purchased at Walmart because I saved some money that way, CVS is ridiculously expensive for everything, I can't believe they're still open, and Rite Aid isn't bad, but Walmart always beats them (I hate shopping at Walmart, I really do to the point I feel guilty, but since I went through cancer and was off work for a year and a half I have been trying to rebuild my savings). If you go to Walmart the sterile saline and sterile water are in the medical supplies aisle in little bottles near the rubbing alcohol and hydrogen peroxide. The distilled water is with the soda and spring water, and the advantage to using distilled water is that you can buy a gallon of it there for $0.88, versus $5 for small bottles of sterile water or sterile saline...
How are you delivering the treatments to your beardie? I guess what I'm asking is #1 are you actually using a nebulizer or just an oil room diffuser, and #2 are you putting him inside of anything that will trap the vapor in and allow him to breathe it fully? Obviously you can't really put a nebulizer mask on a beardie or a bird, or make them put their face down into a nebulizer breathing apparatus, and especially if you're using a little room diffuser you'll need to make sure that your beardie is breathing in as much of the antibiotic vapor as he can, as this really does work very well, but the vapor needs to be delivered directly into the tissue of his respiratory system. I've seen some people hold their beardies over top of a room diffuser directly over the vapor, which would work to a point, but it just seems to me that most of the medicine is escaping that way.
What I did with my male beardie is bought a plastic Rubbermaid or Sterlite container/tub at Walmart, not a massive one but one large enough for him to feel comfortable in, like a medium sized cube with a lid, the same size used as rearing tubs for babies. I think I got my last rearing tubs at Walmart for $7. Then if you're using a room diffuser it's quite easy, you just place the diffuser inside of the tub with your beardie, and then as far as the lid goes (I tried this several ways) I just placed it loosely, not clicked down on top but rather just tested it on top of the tub, and I found that just putting the lid over the tub sideways so that half the top was covered and half was open, and putting the diffuser under the covered side and my beardie on the open side facing the diffuser worked extremely well. He had proper ventilation, but he would be constantly breathing in the vapor that collected on the covered side before it escaped. He actually seemed to enjoy it (as opposed to my parrot), he just sat there and seemed to relax. And using the plastic rearing tub with the lid covering half the tub really worked extremely well to ensure my beardie was directly breathing in the vapor before it dissipated out of the open half of the tub. This is really very important, as I don't think it would be near as effective or as efficient if you just sat your beardie next to a diffuser in an open room. You can even put the diffuser inside his enclosure as long as it has a screen or mesh lid.
Just as an FYI, don't be alarmed if during or directly after each nebulizer treatment your beardie starts coughing and wheezing a lot, my beardie did this every time he had a treatment, and so do my birds. It stops shortly after the treatment is done, and it's normal for the warm vapor to loosen things up in their bronchials, and also, depending on the antibiotic being used, the medication can sometimes cause their bronchials to constrict, which can cause wheezing. It's normal, so don't freak out.
In all seriousness, I'm a huge believer in nebulizer treatments using antibiotics to treat both animals and humans, and every time I've given my parrots a treatment and the 2 weeks I gave my beardie daily treatments I saw marked improvement in their breathing and in their lethargy within 24 hours. Within a week of doing daily treatments most all of their coughing, wheezing is gone, as was most of the mucous in my beardie's mouth. I'm pretty sure I did the treatments to my beardie every day for 2-3 weeks, I don't remember exactly as it was 5-6 years ago, but I also didn't use Gentomycin, but rather Tylosin. So follow whatever dosing instructions your vet gave you for the Gentomycin.
Good luck, and if you have any questions at all just ask, I know Tracie is very familiar with antibiotic nebulizer treatments as well.