According to Jean-Luc himself in the 3rd season Picard episode No Win Scenario, "Jack Crusher was once my very best friend at the Academy." He tells a story about them stealing a shuttle together when they on the Stargazer together, insinuating that they were cadets or ensigns at the time.

But this seems to go against our prior understanding of the Crusher family and Picard’s relationship to them in various TNG episodes.

There are some other issues that make it difficult to resolve this problem: pinning down specific dates in Star Trek is always a mess; the ages of the characters are mostly non-canonical; background images and scripts are generally also considered non-canonical; and the ages of the characters often don't match the ages of the actors.

In the last episode of PIC season 1, Et in Arcadia Ego part 2, Picard says he was 94 years old. PIC is the "present" in Star Trek, more or less, and is about 377 years in the future, so TNG started in 2364 (our 1987). In the same episode Data says he died in 2379 (our 2002, when Nemesis came out). Season 1 of PIC should therefore be 2397 since it aired in 2020.

In TNG’s The First Duty, Picard says he was in the "class of ’27", and since Starfleet Academy is roughly the equivalent of a present-day American university, he was probably 18-22 years old (or maybe 17-21). Assuming he was 22 years old in 2327, he was born in 2305. The personnel files in the TNG episode Conundrum note that Picard was indeed born in 2305 (and on July 13, Patrick Stewart’s actual birthday). Background images are usually considered non-canonical though, unless confirmed in dialogue, as is the case here; aside from his graduation date, his birthplace is listed as La Barre, which is mentioned in TNG Family and various PIC episodes.

So taking 2305 as his birthdate, season 1 of PIC apparently actually takes place in 2399, when he was 94. Close enough to what the date should be.What about the Crushers? As far as I am aware, none of their birthdates or ages are ever mentioned directly in dialogue, except for Wesley, who is supposed to be 15/16 at the beginning of TNG, which puts his birth in 2348 or so. Beverly and Jack Sr.’s ages are not mentioned, although Beverly’s personnel file in Conundrum (again, a non-canonical background graphic) says she was born October 13, 2324, so she was about 24 when Wesley was born.

The only information about Jack Sr.’s age, as far as I know, is that the hologram version of him was supposed to be “about 24” according to the script for Family. The hologram was made when Wesley was 10 weeks old, so presumably also in 2348, and if Jack Sr. was 24 at the time then he was also born in 2324 and was the same age as Beverley. (I assume the script is even less canonical than background graphics, though.) Jack Sr. died when Wesley was 5, so in 2353.

The ages of the actors could be helpful. Wil Wheaton was 15 in TNG season 1, pretty much the same age as Wesley. Gates McFadden was born in 1949 so she turned 38 in 1987; if Beverly was born in 2324 she would have been 40. Doug Wert was born in 1961 so he was 29 when Family aired, slightly older than Jack Sr.’s apparently intended age of 24. Of course Patrick Stewart and Ed Speelers are way off: Stewart was born in 1940 and is 12 years younger than Picard (he was 47 in 1987, Picard was 59 in 2364), and Speelers goes the opposite way, he was about 35 in PIC season 3, much older than whatever age Jack Jr. is supposed to be (apparently about 20, based on when Jean-Luc and Beverly last saw each other). So that actually isn’t very helpful. Characters may be much older or younger than the actors portraying them, and their appearance may or may not match their presumed age.

In TNG Jack Sr. was Picard’s first officer on the Stargazer. They were good friends and Picard developed romantic feelings for Beverly. But Picard was the captain and Jack Sr. was a lieutenant commander, so as far as I understood (watching TNG at the time), they weren’t supposed to be the same age. They had more of a mentor relationship like Picard and Riker.

But then this was retconned in PIC and apparently now Picard and Jack Sr. were cadets together at the Academy, which means they were probably about the same age. Assuming Jack Sr. was also born in 2305, then he was 43 when Wesley was born and 48 when he died. Not really a problem, it just means that 29-year-old Doug Wert was supposed to be portraying a 43-year-old Jack Sr. in Family.

Was Beverly also the same age as them? If she was also born around 2305 then she would have also been 43 when Wesley was born, about 58 in TNG season 1, she and Picard would both be about 76 when Jack Jr. was born, and they would both be about 96 in PIC season 3. On the other hand if she was born in 2324 and was 24 when Wesley was born, she would have been about 57 when Jack Jr. was born and 75 (or 76? 77?) in PIC season 3 (closer to McFadden’s age at the time, 73).

The TNG episode Violations has a flashback to Jack Sr.’s death, and 31-year-old Doug Wert and 43-year-old Gates McFadden aren’t really made up to look any younger than they really were. Well maybe McFadden was a little bit, with her blonde wig…Patrick Stewart also had a fabulous wig but he still appears to be much older than both of them.

Picard’s brother and sister-in-law also seem to have the same issue. Robert Picard was older than Jean-Luc, who was 62 when he visited him in Family. Robert must have been in his mid- to late-60s. Rene was a child in 2367, but I don’t think his specific age is ever givem. We also don’t know how old Marie was. As for the actors, David Tristan Birkin was 13 at the time, Stewart was 50, Jeremy Kemp was 55, and Samantha Eggar was 51. So, also not really helpful. Marie was either very old when she had Rene or the Picard brothers are super attractive to much younger women.

So the possibilities are they are all the same age in universe, Picard and Jack Sr. are the same age and Beverly is younger, or Picard is older and Beverly and Jack Sr. are younger.

Solutions:

- Picard, Beverly, and Jack Sr. are all the same age. Beverly had one baby at age 43 and another at age 76. It’s 400 years in the future! A 76-year-old woman could still have a baby, why not.

- Picard and Jack Sr. are the same age but Beverly is 19 years younger. Beverly is attracted to the much older Picard so maybe she was also attracted to a much older Jack Sr.

- Jack Sr. and Beverly are the same age but Picard is 19 years older, as seemed to be the case in TNG. Beverly had Jack Jr. when she was 57 (not impossible even today).

- Since season 3 Picard is actually an android, he’s an unreliable narrator or his memories are scrambled and he’s wrong.

- Android Picard is just messing with Jack Jr. and the story about him and Jack Sr. as cadets isn’t true (although it's implied that Beverly already told Jack Jr. this same story)

And in the end I really don't have a solution! I know what the actual answer is but it's hard to explain in-universe.