Hi, there are a couple of issues here.
I could find no reference to the company "ConvertMedia" being called "Convert Media". It's a small difference I know, but it's enough to make a difference. Google is able to split words and understand where one word is made up of two, but because these two words as you say refer to a different service, then that is the main cause of the problem.
Google's mission is to return relevant results to the person searching and to do that it tries to fully understand (as much as their current algorithm is able) the **intent **of the search. If someone types in "Convert Media", Google will understand that the person wants to be able to convert some media.
The fact that there are a lot of services that offer that means there is a lot of pages Google can return for that search and as such, it makes providing a result for "ConvertMedia" much less likely unless you have a prominent brand. For example, "hoover" has become a generic name for vacuum cleaners, but if you search for "hoover machine" the hoover.com website is displayed top.
If however, you search for "convert media performance advertising" then that search shows a different intent by the searcher and the ConvertMedia website is returned in the search results.
It's a tricky one, but I would first do some analysis to check where exactly ConvertMedia do rank for "convert media" as that will give you an indication as to how far you need to climb to reach page one. I suspect it will be a fair distance.
The other thing you could do is produce a page on the ConvertMedia site that says something candidly about ConvertMedia does not "convert media", but can be mistaken for that service. Having said that, I wouldn't necessarily recommend that because I think it undermines your brand which is exactly what you want to grow and be known for performance advertising.
I am not sure the above helps much but I hope it explains the challenge you face is ranking better for "convert media".
Peter