Is it possible to get surgery to make my nostrils smaller? (Photo)

A rhinoplasty would be a nice option for you.  It would reduce the dorsal hump and nasal projection and in the process reduce the visibility of your nostrils.  A nostril width reduction may not be necessary after the dorsum and tip are addressed. 

Click on the "Learn more" link, just below my response, or go here:
FacialSurgery.com/RealSelf/RealSelf_brycekathleen.html
I made a computer morph of your nose, and an animation of the morph, to show the changes that could be possible for your nose in truly expert hands.You'll see, perhaps best in the animation, that I lowered the bump on the bridge, and I brought the tip back closer to your face. I made the bridge just the teeniest bit concave, because that's a feminine feature, and your nose doesn't need a huge change. From the front it looks great.
You should understand that the changes I demonstrated in the morph require advanced techniques, techniques that most plastic surgeons cannot handle.  Be sure to read the section in the "Learn more" link on how to stay out of trouble while searching for a rhinoplasty surgeon. You should only have surgery if you are able to make yourself very confident in your surgeon's skills. Read that page very carefully. Most plastic surgeons are quite poor at this operation, and I believe my advice will go a long way toward keeping you safe.
I'd love to hear what you think of the morph. Did it seem to address your main concerns?
Your nose is also a good example of why computer imaging is mandatory in rhinoplasty. You need to know exactly what the surgeon is planning to accomplish -- what features he thinks he can change, and by how much he thinks he can change them. When you see his goals, you'll know whether he has an eye for an attractive nose, and whether he shares your opinion of what constitutes an attractive nose. You'll also know whether the changes he proposes are enough to be meaningful to you, and whether he understands your wishes enough to address all of your priorities. But remember, you're not hiring him for his skills with the computer. The doctor must then show you his before and after photos to prove that he can actually accomplish what he draws on the computer.
In your case in particular, you need to find a surgeon who understands all of the issues I mentioned, and how they fit together, and who can actually accomplish them in surgery. 

Based on your photos-you appear to be good candidate for Rhinoplasty, preferably through a closed technique- without a scar across your columella. In the right hands, I predict you should expect a more elegant and feminine nose.