Okay I actually did some napkin math and while this looks unlikely, it's definitely possible. Math below:

All of Q's trip codes that I observed start with two exclamation points and are followed by a string of 10 letters or numbers. The letters have random capitalization. This means, excluding the exclamation points, which are fixed, there are a total of 62^10 possible trip codes, this number is:

839299365868340224

Pretty big right? But...

Then I did a little research on Trump's twitter to understand the characteristics of his tweets. I'll admit I did not take a large sample. I used his most recent 10 or so tweets to calculate the following:

Average characters per tweet: ~180 (we know Trump likes long tweets, this seems to be about right). There are whitespace characters in the tweets, but every tweet also contains all the letters from his name and all the numbers from the date, so I'll call it even. Percentage of tweets that contain an exclamation point: ~80% (this is also his favorite punctuatory mark. Some tweets contain up to 5 of these!!!!)

Then I looked for some existing statistical data on his twitter account. Some of these analyses claim he tweets up to 10 times per day, however, not all of the twits he tweets are text twits, some tweets are video twats. So let's be generous and say:

Average text tweets per day: 3

Okay first let's calculate the probability that ANY of Trump's tweets contain the characters for ANY of Q's trip codes. We already know there are 62^10 possible trip codes. But how many trip codes can you make from one tweet?

Assuming that all letters and number are equally likely, there would be "180 choose 10" trip codes you could make from the tweet. This is 7.6x10^15, which looks like:

7628275984984380

So for each tweet, you could generate approximately 7.6x10^15/62^10 = 1% of the TOTAL possible trip codes in existence! (remember, however, we are still making a big assumption about the distribution of letters and numbers in the tweet. In actuality the distributions are not even, and look more like: https://en.wikipedia.org/wiki/Letter_frequency)

So combinatorics are really powerful, they can generate really large numbers, just like exponents! So what exactly is the probability that any of Q's trip codes could be built from any of Trump's tweets?

Q seemed to start posting 3 years ago, so I'll round it off and say there were 1000 days of Trump tweets for a total of 3000 tweets. Now, only about 80% of Trump's tweets contain an exclamation point, and the number of Tweets that contain at least two are probably less than that. I'll try to be on the generous side and say 30% of Trump's tweets contain at least two exclamation points. This gives us approximately 1000 candidate text tweets to generate the trip code that Q is using (for now, we can also assume Q only ever used 1 trip code). So each tweet (that has at least two !s) has about a 1% chance of success. With 1000 candidate tweets, we just need to calculate cumulative probability P(X>x) using a success probability of 1%. This number actually comes out to be darn near 100%.

Okay so if the above math is right, it's definitely possible to generate Q's trip code from random POTUS tweets. Actually, over the 3 year period, POTUS would generate Q's trip code 10 times with a probability of about 53%. Now the question is, what is the probability that occurs at the around the same time? For this you'll see the probability decreases quite a bit, since Q's trip code has only been changed a handful of times, I'll say it's 10. So for each of these 10 times, POTUS has to tweet around the same time, and the text has to be able to generate the trip code. Considering only the latter, the probability is about 10%, which is pretty low, but not impossible. For the former, ask yourself if OP would still have posted this Q proof if POTUS' tweets was several hours following Q's post, rather than several seconds, and you have your answer.

Now I would like for people to check my above reasoning and tell me I'm wrong. Especially, if someone wants to recalculate the above using the actual text distributions in POTUS' tweets and check how many times Q has changed their trip code. Thanks!