Is informing the wife about the righteous deeds her husband performs while abroad considered showing off (riyaa')?

Informing your wife about the acts of obedience you perform falls under ostentation (sum’ah), which is forbidden if the intention behind it is self-admiration, seeking benefit, or praise from people. However, if you do not intend the forbidden ostentation, it appears that your action is permissible. This is because informing others about an act of obedience for a legitimate purpose, such as intending for them to emulate it or to compete in good deeds, is permissible.

Hypocrisy (riya’) is performing an action initially for the sake of people’s praise or benefit. Ostentation (sum’ah), on the other hand, is concealing an action for the sake of Allah and then speaking about it to people.