So, I was on a course last week where one of the instructors kept saying sub-SEE-quent.
This isn’t the first time I’ve heard this, but I don’t recall when the other times were, or from whom. This is Ontario, Canada, FWIW.
Is there a really a group of people somewhere where this would be considered the correct pronunciation?
It’s not? I been pronouncing it that way all these years! It follows from SEE-quence so whats the problem?
Is supposed to be sub-SECK-went or something equally silly.
I’ve always said it as SUB-sic-went.
That’s the only way I’ve ever heard it. (Or a close enough approximation. Stress on first syllable, with either a schwa or “short” vowel in the second syllable.
I have heard that pronunciation from a lawyer and, while I could be wrong, I got the impression that it was a legal term that happened to be spelled the same as the English word.
SUB-s<schwa>-kwent or -kwint for me. From California.
Now if I have to say “sequence” and “subsequence”, both of which are common terms in software, they’re pronounced “SEE-kwens” and “sub-SEE-kwens” with a hard s sound at the end, not a z.
I could imagine either the OP’s person being confused between the words “subsequence” and “subsequent”, saying one when meaning the other. Or he’s simply applying the pronounciation of one to the other. Or the OP was mis-hearing which word the instructor was saying and meaning.
The OP mentions his own location, but doesn’t say whether the speaker was a typical Canadian born person of European ancestry, an ethnic minority, a recent immigrant, somebody who barely speaks English, or quelle horruer, a Quebecois. The last are well known butchers of the Queen’s English. 
The instructor was a typical white English Canadian female, with a typical Anglo given name.
I had the course material in front of me, and the word she was pronouncing was what I would call your typical SUB-seh-kwent, as in next, or following.
The thing that I’ve heard twice in a couple weeks weird was seconded. I learned it and always pronounced it as seh-kun-did. Some people use sec-und-ed, just like second with ed tacked on.
But twice lately I’ve heard people pull out some bizarre, drawn-out, effected vowel that exists nowhere else in their speech. Seh-kouhnd-ed as best as I can describe, sounds totally out of place for them, like they turn into Queen Victoria for one random word.
Do you also say con-SEE-quent and con-SEE-quence?
Commonly, in English, in a multisyllabic word the antepenultimate syllable - the second-to-last - is stressed. Donald Trump may preSIDE over the executive branch, but he’s the PRESident of the United States, not the PreSIDent. The legislative branch is embodied in the US CONgress, but the associated adjective is conGRESSional. And then, of course, you have the juDICiary.
There are, of course, many departures from this rule, but in most dialects of English “subsequent” and “consequent” follow the rule.
I’ve always heard it as SUB-sə-kwent, used as an adjective. I’ve never heard sub-SEE-kwent, which if I did, might make sense as a lawyerly-sounding noun.
Likewise, I’ve noticed that the courtroom pronunciation of “defendant” always seems to be dee-fen-DANT (secondary accent on first syllable, primary accent on last syllable, DANT rhymes with “plant”).
You can’t always assume just how closely a derived word follows the base word in either spelling or pronunciation. The word “pronunciation” is a case in point: Derived from “pronounce”, yet note that the “nounc” get reduced to “nunc” in both spelling and pronunciation.
Similarly, I once believed that “maintenance” must be spelled “maintainance” because it obviously is derived from the verb “maintain”. Yet the “tain” gets reduced to “ten” when “ance” is added.