No. Austria uses lex sanguinis (ie. citizenship by descent, so you are the nationality/nationalities that your parents are) and therefore you are the nationality that your parents hold. In my case, my first child was born in Austria to a Briton and a Russian, and therefore had claims to British and Russian.
We did not exercise the claim to Russian, since he would have lost Russian citizenship automatically when my wife naturalised as an Austrian, as Russians don’t allow the retention of Russian citizenship by minors of parents renouncing their Russian citizenship.
Were the child’s mother or father to be Austrian, the child can of course take Austrian citizenship (as was the case of my second and third children, who were Austrian by birth) by descent (durch Abstammung) and are treated as Austrian from birth. To note is that this also applies even if the parent through whom they have a claim to Austrian citizenship has only been naturalised shortly prior to their birth (in my case, I was an Austrian citizen for only 57 days before the twins were born).
In the event of the child’s parent through whom they have the claim to Austrian citizenship (most likely to be the father logically) dies prior to their birth, provided that at the time of death the parent was Austrian they would have a claim to Austrian citizenship.
NB: currently under British law, a child who is only British by descent (ie. my son through me, who was born in Vienna) is unable to pass on his/her British citizenship to his children by descent unless they were to be born on British soil, or in the event that they would be otherwise stateless, were this the case for both parents.