What does the medical abbreviation q mean?

Answer 1

q stands for"Every"
Unfortunately some terms have multiple meanings in medical usage (FOB can be "father of baby", "foot of bed", or "fiberoptic bronchoscopy", so it is tough to say anything definitively. In the US, however, q most often represents the word "every". A notation on a prescription that says q4h would mean "every 4 hours".
it stands for "each" or "every"

q.o.d every other day

q6hrs every 6 hours

q.d. each day (from Latin quaque die)

also Latin "quater"
"Q" or "q" on a prescription is shorthand for the Latin term "quaque" which means "every." So, "qd" or "QD" means "every day" and "q4h" or "Q4H" means "every 4 hours."
Every.

Answer 2

I'm frankly guessing on this, so if there's a doctor or nurse who knows better, please correct it:

It's probably short for "subcutaneous", meaning "under the skin".