The standard is to use integers from 0–11 (actually, 0–e or 0–B as I’ll explain in a second) to represent each of the 12 possible pitch classes.
“Pitch” refers to a particular note: Gb4 or F#2. “Pitch Class” refers to the family of pitches that are enharmonically and/or octave equivalent to each other. For example F# in any octave is the same family as Gb in any octave. In the unlikely event of an Ex, that too is in the same family. To refer to the entire family generally, we have arbitrarily assigned numbers to each one, starting with the pitch class that includes C as 0.
So, any B#s, Cs or Dbbs are members of pitch class 0. Any C#s or Dbs are part of pitch class 1. Any Cxs, Ds or Ebbs are pitch class 2, etc. This means that A#s and Bbs are PC 10 and Bs and Cbs are PC 11, but that can be confusing since it’s often hard to see the difference between 1 0 (meaning, perhaps, a Db followed by a C) and 10 (meaning, perhaps, A#). Instead most analysts refer to 10 and 11 with the symbols t and e, or A and B.
EDIT TO ADD:
As Tim points out in the comments, this system does NOT specify octave, so if you're looking for a way to specify a particular octave without specifying a particular enharmonic spelling, it's more difficult. For the most part, this isn't an issue because of the way interval is discussed in set theory. I'll provide a quick précis of the four ways that interval is discussed, but will leave details for a different question:
1) If I care about the full size of an interval AND its direction, I will refer to the "ordered pitch interval" or opi using number of semitones and + for up and - for down. The distance from middle C up to Db a minor ninth above is +13. The distance from middle C down to B below is -13.
2) If I care about the full size but I DON'T care about direction (which is another way of saying that I don't care which note came first), then I will refer refer to the "unordered pitch interval" or upi using only number of semitones. Both of the examples I referred to above would be upi 13.
3) If I don't care about octave distances, but I do care about the order of notes, then I use "ordered pitch class intervals" or opci. This is probably the hardest one to envision at first. I will count the number of half steps it would take me to get from the first note up to the next note even if the interval went down in the actual music. Going from any C to any Db (that is to say from 0 to 1) would always be opci 1. Going from any Db to any C (i.e. 1 to 0) would always be 11. Going from D to G (2 to 7) would be 5 and G to D (7 to 2) would be 7. The main benefit of using numbers instead of note names (and, importantly, of starting our numbering system with 0 instead of 1) is that intervals can be easily figured out by simple subtraction. Subtract the first number from the second number (mod 12) and you have the opci. My four examples above: 1-0=1; 0-1=-1 which is 11 mod 12; 7-2=5; 2-7=-5 which is 7 mod 12. The mod 12 part may be unfamiliar to you, but it's something you use every time you say that something happening three hours after 11 AM will happen at 2 PM. 11+3 is actually 14 of course, but we treat time like a circle, and roll back to 2.
4) Finally, and most commonly, we can refer to the interval in the most general way possible: the interval class (sometimes called the "unordered pitch class interval", but IC is pithier and has an obvious similarity to "pitch class." In this situation we just want to know the how to get from one note to the other in semitones in the closest way possible. The closest way to get from a Db (1) to a C (0) if we don't care which comes first is 1. All half steps, major sevenths, minor ninths, etc. will belong to this same family of intervals: interval class 1. Whole steps, minor sevenths, major 9ths, etc. are part of IC 2; minor thirds, major sixths etc. are IC 3; and so on. There are only 6 possible intervals classes (unless you want to count unisons and octaves).