no

Season 1 takes place over the course of fifteen hours of the same day. It occurs on an odd day (S1E02), a Friday (S1E15), and in September (S1E12, break room calendar; season 2 is set 10 months ahead in July). I think this places the date of season 1 as September 5th or September 19th. For the purposes of this timeline, season one is treated as occurring in September 2025. The evidence is as follows:

On YouTube, there is a trailer titled “The Pitt | Official Trailer | Max” (link) with a deleted scene where Collins says to Robby that Adamson’s death was “five years ago today”. We know the death occurred in 2020, so that makes it 2025.

In episode 6, when Shelby Adamson, Adamson’s sister, sends a thank you note to the ER staff, her letter also says it has been five years. A transcription of the note: “To the ER Staff of PTMC, As my brother would have done, I think of you all today. Sending this as a token of thanks for all that you do and all that you did for Monty, five short years ago. With love, Shelby Adamson”

Then, we have that patient in episode 10 who gets hit in the eye playing baseball. He says he is 16 and he was born on December 7th, 2008. This would only make it 2024 if it were December, but it is not.

Finally, Noah Wyle said at the FYC panel recently that season 2 is intended to be set on the Fourth of July weekend, on the 250th anniversary of the US, which makes it July 2026.

*When given the age of a character, I give two possible years the character was born to accommodate for the fact that their birthday may not have passed yet. So, for example, Cassie McKay says she is 42. If we treat season one as being set in September 2025, she is at the latest born in September 1983 if her birthday just passed and she just turned 42. However, it may be the case that her birthday has not passed in the calendar year yet (like Santos, who is a Scorpio and therefore has a birthday sometime in late October to mid-November). This means she can at the earliest be born in September 1982 (say, late September), and turn 43 shortly after the events of season one. I use similar logic for events that happen at a certain age (Mohan’s dad’s death when she was 13 - she could have just turned 13 or turned 14 later that year; same with Javadi and the year she began attending college).

**Here is the general timeline of training that I used to make the timeline. (Information concerning the American medical school system is compiled from cursory Google searches and browsing sources such as the National Resident Matching Program [NRMP] and Accreditation Council for Graduate Medical Education [ACGME]. Please be advised that there may be errors and variations may exist. I am not in the medical field, so do correct me!) Following the completion of medical school, new doctors generally begin their residency programs on July 1. The emergency medicine residency program at Pittsburgh Medical Trauma Center is a four-year program: R1 (interns), R2, R3, and R4. I am assuming all of the attendings and residents who we know did/are doing their residencies at PTMC immediately matched after four years of medical school and began their residencies in the emergency department in the July following graduation unless otherwise stated.

3. Touji told Gojo the deal because he was desperate too. He sold Megumi to the Zenin, but deep in the heart he hated them. Touji at the very least loved Megumi mama, so Touji most likely actually didn’t want to sell Megumi to them, because she had told him to take care of Megumi. He was conflicted when he made the deal with Naobito, but back then, he thought that was the only choice he had. He said that he didn't care, but he actually did. It's most likely that he forced himself to not care.

Now that he was on the verge of death because of the Gojo heir, another gamble was open to him because Satoru had asked him of his last words. Yes, it was a gamble since he didn't know that he would meet or even being killed by Satoru.

Since Gojo clan has always had bad blood with the Zenin, he was gambling that Satoru could possibly help him with the Megumi situation. Even if his words is “do whatever you like”, it’s more of an implied plea of help, because nothing good ever comes of “selling off kid”.

4. Satoru had most likely knew how the kids in the Zenin clan treated. He also had basically killed Touji, so it’s only proper to respect his last words. He probably at least wanted to check on Megumi. When he found him, he explained the situation with between his dad and the Zenin clan, then he let Megumi decided for himself. Megumi’s decision was all about Tsumiki. 

Since Tsumiki was non-jujutsushi female, treated badly in the Zenin clan was understatement. This was why finally Gojo took over the deal with the Zenin clan, presumably paying off Megumi’s price to them. Megumi working as a sorcerer and enrolling in jujutsu high is also part of his deal with Satoru to pay off that debt.

A bit ETA: Also Satoru went looking for Megumi because, presumably at this point of time he started to realize that he needs to gather strong allies, to shape and educate them and have them take over the enchelon position of jujutsu world, subsequently changes the system that causes Getou's betrayal. He wanted to see if Megumi has the potential to be a strong sorcerer (from his fanbook page).

5. It’s a tradition of the Great Three Families that the head is the one who has manifested inherited curse technique (souden jutsushiki). This is why Noritoshi became Kamo heir even though he was a mistress’ son.

In the Zenin family, who has several inherited techniques, the strongest one is Ten shadows technique, as it has been implied to be able to go on par with the strongest technique ever; six eyes-and-limitless technique like Satoru. That’s why if someone has manifested Ten Shadows technique, the assumption is they automatically get the position as clan heir. 

6. “If gojo is incapacitated then megumi will become zenin head” is a clause free from Touji’s influence and added by Naobito himself. I would imagine that Naobito, based on their tradition, had wanted Megumi, who has manifested the Zenin’s prized inherited technique, to come into the Zenin clan since he knew about Ten Shadows technique. Also, if he let Megumi goes from the Zenin clan, then he would basically let their prized Ten shadows technique free out of the Zenin clan, who prided themselves on this very technique. It would be a shame for them if later this technique is inherited by different family. But when Touji made the deal, Megumi hadn’t manifested his technique, so Naobito didn’t know. Naobito probably hadn’t know too about the Ten shadow technique when Gojo heir wanted to buy him, so he agreed to sell Megumi to Gojo. There's also possibility that Naobito didn't believe Touji's words when he said his son is talented, since Touji didn't even has a lick of cursed energy. Another possibility is this being Satoru; the strongest in their world; as a person, a clan, and a jujutsushi. No one dared to go against him, both in fight and political influence. More so if it's a matter of money, since we know that Satoru is loaded and not afraid of using money influence.