salamalaikum,
I am glad hearing your good intentions. Insha'Allah i hope your son turns successful in his Tajweed course & also in memorizing the Holy Qura'an (Hifz).
However, I am not very much sure about any private tutor for your son but there is a Islamic Cultural Centre here named 'Fanar' which provides many of such courses to the childrens and also to the Adults.
They are located opposite to Souq Al Waqif (near to the corniche & the Emiri Diwan).
Their contact is;
Hamd Bin Gasem st. cross Hamd El-Kabeer st.
Tel. 1: +974 4250250
Tel. 2: +974 4447444

http://www.qatar-fanar.com/index.php?page=aboutus

Apart from all this, since i can see that you're Masha'Allah a practicing muslim, if you wish to admit your ward (under 4yrs of age) in nursery or play school then do let me know because i know a place where they teach the toddlers according to the Islamic values and is also of English curriculum.