What is the term for an integer one larger than a given integer?

The number that's one greater than a given natural number n is called the successor of n. See the Peano axioms, which are a standard foundational definition of the set of natural numbers.

This terminology strictly is only defined for natural numbers, but I think it can be safely extended to integers without confusion.

If it's clear you're talking about integers, then you could say successive if you don't simply want to say next.

But I think the word next clearly implies no intervening values in relation to integers.

You could say "the next consecutive integer".

"The increment" would be technically correct, but it really depends on the structure of the sequence, the context in which you are writing/speaking, and the assumptions that can safely be made about the audience's understanding.

If the structure can be unambiguously understood as an ordinal sequence (such as the index of an array or a series), then you can use words like increment or next the same way you'd be free to use the subscript n+1 in a formulaic description or ++i in a computer program. If the description of the series/sequence is not apparent to your audience (or there is a chance that they might confuse the index/subscript with its associated value) then you can initially define the increment and then use terms like next, subsequent and so forth.