What authentication method are you using? Unless you are using
Integrated or Forms-based authentication, no, the second prompt cannot
be avoided without some other intermediate proxy that exposes
authentication to the client  using one of those 2 methods.
Authentication is scope per-process on the client side for security
reasons, so an authentication session in the browser cannot be shared
with an authentication session in Word OOTB.

if you are using windows authentication you cancheck if automatic logon with
user name and password is enabled in th intranet security zone. or you can make
the site  trusted and in the trusted zonze you can enable this option.