Yes, as long as the college you take it at is accredited. When you apply through AMCAS or ACOMAS, you will need to send a separate transcript from every post high school institution you have taken coursework at.
So you’ll need a transcript from Ochem institution, your quarter institution, and your semester institution. (And any others where you may have earned college-level credits. Such as any dual enrollment classes you took during HS or any CC credits you took at any time during HS or college.)
I see several potential issues with your plan–
Med school adcomms are likely to look askance at Ochem classes taken at a university other than one you attended full time for the rest of your credits. Ochem is a traditional “gatekeeping” class for pre-meds and adcoms will wonder why you took it elsewhere instead of at your home college. They may suspect you are trying to take the class at an easier school. “Game the system.”
A second issue is that taking 2 semesters worth of Ochem in 8 weeks is going to be very challenging. The material will unlike anything you’ve seen before and the pace will be very fast. One poor quiz grade or couple of missed lectures and your grade may be irrecoverable.
One other downside of Ochem over the summer is that due to the compressed time frame, the course content often gets truncated. (Topics that would normally be covered by a regular semester-long class will get glossed over or omitted altogether.)
Lastly, does your new college offer health profession committee letters to those applying to med school? If it does, you’ll need to check with the HP office to see if they will write you a letter if you don’t have Ochem on your transcript. They probably won’t. Not having a committee letter from your college’s HP committee if it offers one is a red flag on your med school application.
Will you also be taking Ochem labs over the summer? The labs are extremely time-consuming.
Since you will sending separate transcripts from every institution you’ve attended, adcomms will be able to see that you’ve completed a full year of gen chem.
The bigger question is: since your new college only gives you 8 credits for the class, not the 9 you’d get with 3 quarters, will it be enough to let you fulfill graduation requirements at your new school?
Will it allow you to satisfy the pre-requisites for any other UL level chem classes you will be taking? (Like biochemistry…)
Which also brings up the question….if your new college doesn’t accept your summer credits for OChem, will you be able to take biochem if the pre-req credits for Ochem don’t appear on your transcript?
I strongly recommend that you speak with someone from the chem or bio department about the pre-req issue. (Chem if biochemistry is offered through Chem Dept; bio if biochemistry is offer through the Bio Dept.) Your college registrar has the final say on whether you will be able to enroll in biochem so make sure you get advice directly from your new college’s advisors.