Is it known how Hilbert initially reacted to Gödel's incompleteness theorems upon their announcement at the Königsberg conference in 1930, or their publication in 1931?
The paper 'Some facts about Kurt Gödel' by Wang (1981) (regrettably paywalled) contains a section that suggests Hilbert was not present when Gödel originally announced his sketch of the First Incompleteness Theorem at Königsberg, on the 7th of September, 1930.
Notable mathematicians that were present include Carnap, Heyting and most importantly von Neumann, who shortly thereafter managed to independently prove the Second Incompleteness Theorem, but decided to leave the credit for its discovery to Gödel, after the latter informed von Neumann that a paper containing both important theorems was already in the process of being published.
From the link provided by Philipp in the comments, it is clear that Hilbert reacted angrily when the paper by Gödel was published, since it meant the failure of his program. However, being a mathematician, he could not argue with the validity of the proof and therefore resigned himself to the truth eventually.
Constance Reid's biography Hilbert contains a brief discussion at the beginning of chapter 23. Hilbert was
somewhat angry... but then began to try to deal constructively with the problem... Broadened methods would permit the loosening of the requirements of formalizing. Hilbert himself now took a step in this direction. This was the replacing of the schema of complete induction by a stronger rule called "transfinite induction". In 1931 two papers in the new direction appeared.
I think the last sentence may refer to the consistency proof for Peano arithmetic due to Gerhard Gentzen, who was Hilbert's assistant. Gödel himself gave a consistency proof for Peano arithmetic using so-called functionals of higher types (see Shoenfeld's Mathematical Logic).