The answer to your question is going to be very institution specific. First, ignoring the visa issue: In general, there's nothing that prevents a resident from getting an honorarium from their institution, or another. Presumably getting this honorarium requires your presence / time at another institution, and your home institution needs to approve that. Rarely, an institution could try to claim that, if you're presenting work you did there, that they own the IP of your work and hence they get to claim the honorarium. The amount is usually small enough that it's no worth anyone's time/energy to do so.
The J visa issue is more complicated. What's clear is that you can't just get paid by someone else if you're on a J visa -- you need approval from your sponsor. And your J-1 sponsor for a GME position is NOT your home institution -- it's the ECFMG. Whether they will be willing to do the paperwork to approve this is unclear.
So, best plan is to discuss with your program. If they are in support (which hopefully they would be), they can help you petition the ECFMG. Whatever you do, don't try to collect it without ECFMG approval -- if they hear back that you did so, you'd be in violation of your visa stipulations and could have your visa revoked. If you can't get ECFMG approval, you can still do this but would need to decline the payment.
