Is my prayer in my father-in-law's mosque, due to his observation of me and my desire to reassure his heart that his daughter's husband prays, considered showing off (riyaa')?

If your original intention behind performing prayer in the mosque was to draw closer to Allah, and the other matter was secondary, then that does no harm and is not considered ostentation (riya'), especially if the immediate gratification of the self occurs as a consequence of the act of worship and was not intended, or if it is something that helps one increase in worship.