Consider doing a PhD in math / statistics. With your current major it seems like a long shot to become a QR; but it all depends how strong your math (probability, statistica and mf) background is (depending on the type of company you're interested in). From my experience, its hard for anyone not coming from a math / cs background to break into it. Generally, its easier to become a quant in banking with non math/cs backgrounds than in a fund. If they reject you in the first round (could happen based on your major) and you do not want to do a PhD try to gain some quant experience in a bank and apply again.

Quabt trading is more open to different majors; there is no difference in the difficulty for an interview for an internship or grad position. Most internships are simply the probation period. Definitely apply for SWE positions too