I am getting authentication prompt while opening office documents from Document Library? How to avoid this?
I don't want to change any settings in browser, since it is not possible to change in client side.
There are numerous scenarios that can lead to user's getting prompted. The Authentication providers, your host header/domain name, etc all play into it.
The most common deployment scenario is a claims setup with Windows authentication. If you are using a host header/fully qualified domain name that doesn't match your AD domain, then it will not automatically relate the system to your network or the Intranet Zone in IE. So users are prompted.
Adding a Trusted Sites, or Intranet Sites entry would resolve for those users, but would require either manually making the setting change or you can push it out with a Group Policy.
On both FBA and AD (Claims), Office will need it's own authentication ticket as well as the browsers. That likely also explains why your same user has different behaviour on different machines etc.
You need to get users to accept they must login both through the browser normally and if they are then opening documents, to also login again for Office - however they will only need to do that second login (rather irritating, admittedly!) once - Office will remember, so the second login will not be needed going forwards on future days/weeks. These auth tickets are subject to cookie settings so you can control them through that.
I think this is your problem - just do the second Office login (on each machine unfortunately) and it should be sorted for each user.