Antimatter is a consequence of quantum field theory, but all signs indicate gravity can not be successfully described as a quantum field. Gravity has properties such as having its interaction strength grow with the amount of charge (energy), which is very different from the forces described by quantum fields that have constant interaction strength. To get antigravity, you need to find a way to have negative energy without it making the vacuum unstable.

It's most typical to describe gravity as dynamical spacetime, not a field existing in that spacetime. The closest people have come to a quantum theory of gravity has been to show that in certain toy universes gravity in a bulk space is the dual of strongly-coupled field interactions existing on the boundary of that space-- that is holography. Information-theoretic approaches have have also suggested that gravity emerges as the 'hydrodynamics of entanglement.' All this suggests gravity is not a fundamental force, but is far from a coherent real-world theory at this point.

The closest to an "antigravity" in general relativity is the cosmological constant, thought to explain dark energy. Its tiny (10^-122) positive value behaves like a constant negative energy attributed to the vacuum itself, driving the exponential expansion of space.