If you know all the major key signatures, you can use them to identify intervals. Example:
I see the interval A-C, and recognize that it's a third. I know that the key of A has a C# in it, so a major third up from A is C#. Since our interval is a half step smaller, this must be a minor third.
The only way this doesn't work is if the bottom note doesn't have a key signature (for example, A#). In this case you should remove the accidental from the bottom note and change the top note accordingly. For example, if I had the interval A# to CX, I would lower both notes a half step, ending up with A and C#, which we know is a major third. With a bit of practice, you'll start recognizing intervals without going through the whole process.
