So I'm a recent college graduate and I was accepted into the Engineering I Co-op program for the upcoming September. In my admission offer it said that they approved 6 units of transfer credit (two first year courses).

I understand that first years in engineering have to register 37 units of courses but since I've been granted transfer credit for two courses, does that mean that I am exempt from having to register into those courses? Will SOLAR tell me if I only need to register 31 units on registration day?

Yes, if you received transfer credit for 2 courses, then you do not need to register for the courses on SOLAR. In fact, SOLAR will not let you register for those specific courses. To move on to level 2 engineering, you must complete 31 units of the mandatory engineering courses - you can move on without having to complete 6 units of complementary studies electives. So, if you decide that you do not want to take any courses in the place of the transfer credits you received, you need to register in only the remaining Engineering 1 courses.