I have a new trigger, a Yongnuo brand RF603N II and a flash, Godox brand V860 II. Both devices work well, but the trigger does not have any effect on the flash. Does anyone have any idea why or how I can connect the trigger to the flash otherwise?
Yongnuo's radio triggering is incompatible with Godox's. If you do not have a second Yongnuo RF-603 II attached to the foot of the V860II-N to act as a radio receiver, they can't "talk" to each other.
If you do have a second RF-603 II and it's not working, check that your transmitter unit is seated fully forward in the hotshoe; in TX mode not TRX mode if it's not being used on a Nikon or Canon camera; that both are set to use the same channel; and that the V860II is in 'on-camera' mode (see also: My Godox flash won't fire off-camera. What should I check?). But the big drawback of using RF-603 II triggers is that they're manual only and all you can tell the flash to do is fire.
Most of us would get a Godox transmitter, because it could use the built-in transceiver in the Godox V860II, eliminating the need for an attached receiver unit. It would also let you use TTL and HSS, and give you remote power, group, and zoom control over the flash. I would recommend looking at a Godox X2T-N or XPro-N; the Adorama exclusive Flashpoint R2 Pro II-N; or another Godox TTL speedlite (like a TT350) as your trigger, not a Yongnuo RF-603 II. If you really are stone broke and in the US, and only want to pay $25 for a trigger, there's also the Flashpoint SPT transceiver, but it's more like a Yongnuo RF-605. You'll only have sync (fire) and group control.