Linguists don’t like the word axioms. as you can tell from the other answers: they imply a degree of mathematical rigour that just isn’t compatible with someone as messy as human language. But there are foundational assumptions to disciplines in linguistics, which are pretty much axioms. And they would be more overtly acknowledged, were linguists a more reflective bunch.
In historical linguistics, for example, you have the Uniformitarian principle. (I see from Wikipedia that Uniformitarianism is actually a geology thing, and you can see why linguists made the connection.) That’s the assumption that languages worked the same way 5000 years ago as they do now. You need that assumption to do any non-trivial historical linguistics: our documented processes of language change reach back only 2 or 3 millennia, and we need to assume that the same processes worked further back in time, if we want to say anything at all about Proto–Indo-European.
In generative syntax, you have configurationality, the principle that you can build up the syntax of a language out of phrase structure grammars (the kinds of grammar computer programmers are used to). Non-configurational language are a shock to generative syntax: they are languages where word order is seemingly random, and words do not hang together in well-defined phrases; so you can’t write a simple grammar (the kinds of grammar computer programmers are used to) to account for those languages’ syntax.
There’s other foundational beliefs, such as the primacy of spoken over written language, the “natural” evolution of language, the exceptionlessness of language change, the unidirectionality of grammaticalisation, the arbitrariness of the sign, and so forth.