Last week in my AP music theory class my teacher set up an exercise before class and asked is to write the following diatonic chords without a key signature G maj triads, Fmin triads, Bb major 7th chords, and Bmin 7th chords. Here is the part I don't understand, my teacher said that since there is no key signature the minor diatonic chords become harmonic minor. I have no clue why this is and I'm still confused. He just said if it asks for minor assume it's harmonic minor. Can anyone see why this is?
If there is no key signature, the exercise is just about "writing chords" not "writing chords in a given key".
For a simple minor chord, it makes no difference what version of the minor scale you "assume".
For 7th chords, the interval of a major 7th is always one semitone less than an octave, and a minor 7th two semitones less. A dominant 7th chord always has a minor 7th.
There is no deep reason for any of this - it's just how the different chord names are defined.
I don't know if an AP course would cover more complicated chords like 9ths, 11ths, 13ths, added 6ths, suspended 2nd and 4ths, chromatically altered notes, etc - but I would guess not, because the "complete" form of many of those chords contains more than four notes, so you also have to consider which notes of the chords should be omitted in four-part harmony.
As far as I can tell, the "assume harmonic minor" part applies only to the minor 7th chords (the rest of the chords shouldn't change regardless of whether you're in a major or any minor key)--but it's important there.
This will no doubt sound ugly, but based on the "assume harmonic minor" note, I'd write the B minor 7th chord as B-D-F#-A#. Yes, it's Bm maj7, but it uses the 7th scale degree of B harmonic minor.
If that's not the chord the teacher asked for (or if your teacher marks that as wrong), ask for clarification.
I think it's a little unclear what is being asked of you based on the way you worded it. When you say that you were asked to write out the G major triads, I'm understanding this to mean that you were asked to write out all of the triads found within the key of G major but I'm not sure everyone else is thinking of it this way, where some comments have suggested that you're just writing a single triad or 7th chord per example. I'm guessing that the idea of this exercise is to have the students demonstrate a more thorough understanding of chords within a given key by not relying on a key signature, ie, you have to know which notes would have the accidentals that the key signature would usually allow you to not have to think about.
The assumption that it is harmonic minor is probably based on the idea of functional harmony in general. Typically if you are writing music using functional harmony, the V chord will always be major/dominant and the vii chord will always be diminished. I'm not sure of what the teacher truly wants but if I were coming up with this sort of exercise for students of mine, I would be trying to get the below answers for 7th chords in G minor.
G min 7, A min 7 b 5 (A half diminished), Bb major 7, C min 7, D7, Eb major 7, F# diminished 7.
So I'd be trying to have the students write out these chords to show understanding of how they typically appear, where you wouldn't usually end up having G min/major 7 or Bb augmented/major 7 appear within a piece and think of it as a diatonic chord.
It may be possible that the teacher wants you to provide these less than common chords that are not exactly thought of as diatonic but not what I'd be thinking of. I would suggest asking the teacher for clarification. All too often in my previous experiences as a student, I found that my fellow students wouldn't ask the teacher for clarification and essentially struggled (at times) as a result of this. I was always one to ask lots of questions and volunteer to answer questions in class. This allowed my to one, always know exactly what the teacher wanted (which often helped my classmates that didn't bother to do so), and two, when answering questions, I was getting confirmation that I understood what was being taught. In your situation, I would not only ask for clarification on whether or not harmonic minor is to be used for all chords or just those that need it for functional reasons, as well as why, particularly asking why if I was being asked to use harmonic minor for all chords, since those other chords don't have a standard use in functional harmony. This isn't to say that those other chords aren't used at all, just that they are incredibly uncommon in most types of music. Classical music did venture into some very weird stuff but it was starting to break outside of functional harmony when it did so. Min/major 7 chords do exist in the wild, as well as augmented chords, but I wouldn't think that this would be part of an exercise in writing diatonic triads/7th chords.