If you live in or neare Gharrafa, you could check out Giggles day care and nursery. They are located in Nasiriyya (near the Education City housing lots and LuLu Express) and have at least 2 facilities quite near each other as far as I know. Don't have any experience with them, though. We did visit the nursery once some 3 years ago when we were considering one for our then 3 year old. They do offer to care for children (including infants) from morning till late afternoon (3:30 pm).

I'm sure there are other nurseries and day cares around Doha that offer similar services.

Do visit the nursery section on Doha Mums site for more information.