There are several web design companies in Nigeria especially lagos. Sadly most of them offer very poor services as they know nothing about website building.
I have done my research and here's what i found out. The best solution providers are not those very big names, small companies with great minds are making names.

Name of company:Touchcore limited
website url: https://touchcoreltd.com/
location: 14B Wole Ariyo, Lekki Phase 1
contact phone number:01-291 8381,08170000560

NB: This list is a pointer to web design companies in lagos who give you value for your money. Feel free to add yours.