"Morality is inborn" can be read a few different ways. For example, it could mean that our capacity or tendency to form moral beliefs (beliefs in propositions of the form "it's morally wrong/morally right/permissible/obligatory/ought to be that P") is inborn. This says nothing about the veracity of any or all moral beliefs, or that there even is an objective morality which such beliefs are about; even a moral nihilist or an error theorist could say that we have such an inborn capacity or tendency, but that such beliefs have no grounds, are all false, are all meaningless, are relative to culture, and so on.
I think the way you are intending "morality is inborn" to be read is something like the statement of ethical intuitionism, the position that some moral truths are self-evident, or known by us in a direct, non-inferential way. Since these are moral truths, they are unlike how I described in the first paragraph; the innate beliefs are knowledge and really do correspond to some moral reality. The ethical intuitionist might claim that we "see" the correctness of certain basic moral beliefs, analogously to how we "see" the correctness of basic mathematical beliefs. Non-basic moral beliefs are built around the basic beliefs like how we build mathematics around our basic mathematical beliefs, so that our theories cannot contradict these basic beliefs. If we deduce from our mathematical theory that 2+2=5, we reject it as false (or at least not useful for some purpose at hand; there are interpretations of the symbols in "2+2=5" where it comes out true, but not in "normal everyday" math). Likewise, if we deduce from our ethical theory that gratuitous harm is good, we reject that ethical theory as false. We seem to appeal to these supposed basic moral beliefs when arguing for and against ethical statements all the time.
If something like the above is what you mean, then looking at the SEP and Wikipedia pages for ethical intuitionism there are at least three objections:
There is the objection that states that there is no objective morality and no objective moral truths, and so there is nothing to know. It follows that there cannot be innate knowledge (since knowledge implies truth) or some faculty that is supposed to perceive such objective moral truths. The faculty is, at best, like how I described in the first paragraph. A materialist might reasonably make this objection if they thought that there is no way to ground objective morality in material reality.
There is the argument from disagreement. The innateness of morality seems to run contrary to the fact that we disagree wildly about what supposed basic moral beliefs are supposed to be. This is unlike the case of basic mathematical beliefs, or basic beliefs about sense perception since in those cases we usually all agree. And in cases where there is disagreement about basic mathematical or sense perception beliefs it's usually quite easy to figure out why (we were looking at the object of sense perception in funny lighting, a different angle, were under the influence of mind-altering drugs, etc.). Ethical intuitionists owe us an account of moral knowledge that can explain this disagreement.
There is the "Occam's razor" or "no good arguments" objection that denies we need to posit the innateness of morality since it explains nothing. Psychology, sociology, and neuroscience explain away the experience of morality, rightness, wrongness, etc. without needing to posit some innate moral faculty or knowledge. Since we don't need to posit such a faculty for explanatory reasons, ethical intuitionists must provide some arguments for why they believe there is such innate knowledge.
Of course, there are responses to each of these objections, and responses to the responses, and so on... You can check out the bibliography on the SEP page for more in-depth reading material!