Well you have to use iether a Prospector or a MOLE, since Quantanium is not minable by ROC or hand mining equipment.
Also, regardless of the ship you'll choose, you'll need a new minig head, sice the stock one will caouse a lot of headhacke: you can use the Helix (cost around 100k) or the Lancet (corst around 25k) both are good enough for dealing with quantanium deposit.
Best mining location for quantanium is the Aaron halo. you can travel there by traveling to of from Microtech and break off quantum travel at the right moment. Keep in mind this location is really crowded and a good spot for pirate to fish for pray.
One mined remenber that is a un-stable material and will degradate over time, so be use to have a good Qauntum drive to allow you to travel fast toward MIC-L1 (or the nearest L1 station) for refing. pro tip: as you land store your ship by ASOP terminal, this will stopt the count down of the degradation and will allow you all the time you need to get to the refinery.
For refining the most common used metod is DInix Solvation: it's slow as hell having a required time around 28h, but it give you a good yeld and doen't cost a leg and a kidney.
For selling best place is Area 18 TDD since it has a higter selling cap and regeneration (that is true for all the matirial btw...).
A war do of advice: when you retrive the refined material be sure to be able to store all of it: all the extra goods that can be fitted in the cargo are lost!