I've been experiencing the same issue except I've had to reload the page each time I complete a quest. Simply closing and reopening the quest window does not move the quest line forward. The quest line locks up after I've collected the quest reward, and it does not advance to the next quest until after I refresh the page. Instead, there appears to be no quest in the quest line. However, everything else about the quest itself seems to keep functioning. For instance, if I collect coins before reloading the page and the subsequent quest was "collect xx amount of coins," I do still receive credit toward that quest from the coins collected prior to reloading the page. It's just the quest line itself that remains empty until the page is refreshed.
This problem occurs both in the main game quest line, as well as the quest line for the settlements. And if it matters at all, I use the Chromium browser on an Intel Nuc running Ubuntu Studio.