I cannot really comment on Chrome, so I'll just quote them.
Extensions installed by third party programs using external extension deployment options will be disabled by default. When a third party program installs an extension, the Chrome menu will be badged, and users can click through the Chrome menu to see a dialog containing an option to enable the extension or to remove it from their computer. (Source: http://blog.chromium.org/2012/12/no-more-silent-extension-installs.html)
Not sure if the "badge" bits are still the case, though, considering this post is from 2012 and IIRC there are plans to outright disable all non-webstore installs at some point.
Regarding Firefox: No, there is no supported way of doing silent or auto-enabled add-on installs in Firefox. In fact, attempting to circumvent the opt-in dialog violates the Add-on Guidelines and can get your add-on blocklisted, even if your add-on was never listed on the official add-ons site.
Regarding @Noitidart's comment:
I also don't know how to do silent install but this guy says in a comment that addon sdk does it silenty, so check it out: https://ask.mozilla.org/question/301/open-firefox-profile-with-nsiprocess-and-access-it-via-privealaged-scope/?answer=311#post-id-311
Well, the cfx run
command of the SDK overwrites some preferences in the Firefox profile (it usually creates) to have the add-on auto-enabled. This is OK for development stuff, but is not OK if you push your add-on to unsuspecting end-users and therefore is a blocklist-able offense.