Can you store luggage for long periods of time (1 week) at Tribhuvan International Airport in Nepal?

What would a traveler recommend for large suitcase that I dont want to take with me on a trek from Kathmandu?

Hello,
According to your question, it seems that you have a plan of some days to spend in Nepal. It is obvious that you will have to stay in one of the hotels in Kathmandu. I strongly recommend you to store your luggage in that hotel coz I think it won't be safe in the airport. Your luggage will be very safe there. They are very responsible in any service they offer.

sujan from kathmandu

as far as i know its better to carry out your luggage the very day when you arrive Tribhuvan Int'l Airport.
since you are going for a trek from Katmandu, i think you will be staying there in hotel/guest house so before leaving for trek, you can tell your hotel manager and leave your belongings there.
-- but i think manager would like to know what's in the box.

No you cant store it at Airport
Any hotel or your agent may keep it with a minimum cost, this is safe (dont forget to take slip if in hotel)