You need to call them, but generally bear in mind:

*Q class is a rather heavily discounted economy fare ticket.
*Q class cannot be upgraded with miles. Only Y, B, H, K and M. These are higher price / less restrictive economy tickets.
*Even if you were on Y, B, H, K or M, you cannot upgrade to J with miles. PEY only. (One class upgrade allowed max.)

If you want to pay cash, it will be tricky but likely feasible. The bottom line is Q fare likely comes with onerous rebooking and cancellation fees. But, if you really want to go to J with cash, I've found them amenable to these types of situations in the past, provided the cash outlay is significant. In your case, to upgrade to discount J inventory one way will be harder than round trip, and they may only be able to quote you the full J fare if you want one way. This will likely be a multi-thousand USD fare difference fyi.

There isn't some "can I pay a few hundred bucks" to upgrade my ticket thing, unfortunately if that's implicit in your question. At best you will be able to pay the fare difference, which will likely come with a hefty price tag. And just a word of warning there is a chance they won't let you change the ticket without paying the penalties, since it's possible that Q fare is nonrefundable or non-changeable or something like that.

This probably isn't what you wanted to hear but just trying to be realistic.