Is it possible to enter password for sudo only once, and configure it to not require for password on other terminals?

Sure it is. Run sudo visudo and add this line to your sudoers file:

Defaults        !tty_tickets

As explained in man sudoers:

 tty_tickets       If set, users must authenticate on a per-tty basis.
                   With this flag enabled, sudo will use a separate record
                   in the time stamp file for each tty.  If disabled, a
                   single record is used for all login sessions.  This
                   flag is on by default.

By setting tty_tickets to off (that's what the ! means), you enable a single authentication to be shared by multiple sessions.