Error opening https sites from home?

[DunbarPappy®ϟϟ]

Most common problem:
Right click tray time> "Adjust..."> Update.
While here, tick 'Automatic synchronize...'

I don't believe it's related to the function or settings of the router; that is a straight forward device that does not examine 'Certificates' or packet contents for that matter: only headers.
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Taking a different tack, you might be encountering "3rd party assets" (remote sources, usually advertising) that have a Certificate issued, but it does not come from the 1st party (the site in the address bar) and so Firefox is flagging it, for a reason I'm not sure of just now.

Try making these tweaks and see if it wok relieve some of the problems:
FIREFOX: Tools> Options> Privacy> top drop slot set to "Firefox will use custom settings..."> Check 'Accept Cookies from sites'
â–ºUn-Check 'Accept 3rd Party cookies'.

A good collection of effective security & privacy add-on's (Official Mozilla) is here:
https://addons.mozilla.org/collections/d...nbarpappy/
"AdBlock Plus" (with 'subscription' added) helps defray ad rubbish.

(From that collection, also get "Perspectives" (a Certificate Notary service) and "Certificate Patrol" (monitors and notifies of Certificate changes and essentially interrupts the 'Trust on first use' behavior of browsers until you can look it over and accept or deny it).