Yes, India Operates some active International passenger train service connecting with neighbouring countries like Nepal.
Jaynagar–Kurtha / Bijalpura Link (Nepal) Demu Train.
Operates passenger train services connecting Jaynagar in India to Nepal (visa-free for eligible citizens)
It Connects Jaynagar in Bihar, India, to Kurtha and Bhangaha/Bijalpura via Janakpurdham in Nepal, Travel duration is approx 2 hours and 33 km distance. 2 service in a day.
It is operated by the Nepal Railway Company, utilizing Diesel Electric Multiple Units (DEMUs) supplied by India (Konkan Railway).
It was inaugurated on 2 April 2022 by Indian Prime Minister Narendra Modi and Nepali Prime Minister Sher Bahadur Deuba, and regular commercial passenger runs officially began on 3 April 2022.
Currently this is only international train operating from India and rest of the services towards Bangladesh and Pakistan were discontinued, and those services aren't started yet.
Source:- India-Nepal train services begin today; Here's all you need to know | Today News