Apart from a single, incessantly shared "screenshot" that could have been easily fabricated, there is no evidence that such a post exists, or ever existed. According to a Trump spokesperson, the screenshot is not real.
Fact Check
On Sept. 9, 2022, one day after it was announced that Queen Elizabeth II had died, social media erupted with chatter to the effect that former U.S. President Donald Trump's had claimed the queen "knighted me in private."
The chatter revolved around a post Trump had supposedly made on his own social media platform, Truth Social. A putative screenshot of the post was met with both credulity and skepticism, mostly the latter. (Nota bene: Foreign nationals are only eligible for honorary knighthoods in the U.K.)
No evidence exists whatsoever that Trump posted such a message on Truth Social. No such post currently exists, nor are there any signs (in the form of shares, questions, or chatter) that it ever existed. Moreover, every "screenshot" of the supposed post we were able to find on Twitter, Facebook, and other platforms was exactly the same -- only one such "screenshot" exists. It would have taken less than a minute for someone to fabricate it.
There are no archived versions of the alleged post to be found, either. What one does find, regarding the queen's death, in the Wayback Machine's archive of Trump's Truth Social account are posts expressing his and Melania's respect and condolences to the royal family and the people of the U.K.
Finally, we note that The Associated Press reported receiving a message from Liz Harrington, one of Trump's spokespeople, confirming that the post was fake.