I'm having the same problem, despite launching it via GOG Galaxy, the game never connects to the server. I've played for a few days now and am yet to see my first quicksilver mission.

I've tried disconnecting and reconnecting to the internet while the game was running, but it didn't work.

When 3.02 came out on the 29th, for the first time I stayed connected to discovery services the entire time I played which was roughly four hours. Unfortunately, since then I have to disconnect my internet connection on the PC and reconnect it to get a short burst of connectivity that lasts anywhere from a couple of seconds to a minute or two. Hopefully they can get it working for good again sometime soon.

Dont know if this will be any help to those who cant initiate their quicksilver missions, but from my experience you have to reconnect to the internet right after disconnecting (while the game is running), i.e. hit disconnect and connect right after the button becomes available.

Looking closely at the discovery page, it seems to, obviously in error, report online discovery services being back online at the instant the internet is disconnected! Then once the connection is re-established, it either stays green for a few momments, goes blue (connecting) or red (connection lost) right away.

Because the duration of this burst of connectivity is random and potentially very brief, as others have mentioned, it is best to already be standing in front of the Nexus before attempting the trick.

TL;DR:
1) Stand in front of the Nexus.
2) Alt-Tab and disconnect from the internet.
3) Re-connect as soon as the button becomes available.
4) Alt-Tab back to the game and wait for the QS mission to pop-up.

Same problem, can´t connect to online discovery service. I have a lot of things to register

I can do quicksilver missions but i have no connection to the online discovery . Any news on when this might get fixed?

So... one week and it still is busted. Had connection once after described Internet connection reset - for about 30 seconds. Not very impressive. No Info about when this will be fixed. Nothing on Steam either.
Edit:Sent a ticket per Zendesk page. Everybody with this problem should do. Maybe devs will listen then.

As of yesterday i have discovery connection now. No idea how long it will last but so far its been up +24 hours.

Same for me, both yesterday and today i 've been experiencing constant online services connectivity.

Yes, today the game updated and online services are back working for me.

Same here, finally I have online discoveries

I've had the game for four weeks and I've never checked and found it connected. Systems that I've named appear with the original names for days in my teleporter list. I bought this game a second time for the multiplayer, and it does not work at all. I can't even enter friend codes.