The 68' movement consisted of students and the occupy movement seems to mainly consist of people younger than 40 (source).
I'm wondering if there is an age group that is more likely to protest the status quo and why.
If you define "conservative" as supporting the status quo, then conservatives are the largest idological group. Only people aged 18-29 are equally as likely to be conservative or liberal, where all other age groups a person is twice as likely to identify as conservative. So, people 18-29 are more likely to protest the status quo.
It is commonly believed that as people grow up they become more conservative, which is reasonable considering the fact that those '68 protesters of the status quo would now be in their 60s-70s.
"A young person who is not liberal has no heart; an old person who is not conservative has no brain."
(Your choice of attributions for that quote.)
Younger adults feel more invincible and take more risks; this is wired in as it offered an evolutionary advantage. Decisions also tend to be driven more by emotion and less by rational reasoning for younger people; the decision to protest is often one driven by emotion. Risks are associated with growth and young people are going through a lot of that. They have more years in front of them and therefore more to gain if they get the change they demand, because they'll likely be living under it longer than the older folks. They haven't grown as accustomed to the status quo and don't have as much invested in it already. And in their lives, they see big changes around them, and they believe that big changes are still possible, if you work hard and try hard for them.
There is sort of a minimum age for having the freedom, permission, agency, and means to be able to attend a protest independently, as well as a sufficient awareness of what's going on, ability to understand that in some relevant context, and sufficient level of caring about it to motivate protest.
Older folks have tried and life beat them down. They're less likely or excited to try again because of the memories of those past negative outcomes. They have more experiences telling them that fighting for change is hard and likely to beat them down, and more experiences that the status quo is not necessarily that bad especially compared to the cost and likelihood of change resulting from action (which younger people probably overestimate and older people probably underestimate). Some of those who don't and actively press forward for change may also have more ability to influence change through formal channels (and therefore less need to protest), that younger people lack.
So in conclusion, young adults are the age group most "likely to participate in social change/protesting movements/actions."