Is project manager a non billable role in Cognizant? Will it effect my job security?

It is billable

It's a billable role if you are associated with any project. It's better to be in a project so that the billability starts. It would not impact your job security initially on a shorter time frame (say 1-2 months) but if you are on bench for a longer period of time it will certainly effect and may lead to termination. So try to get into a project / account, learn new skills and get a absorbed to secure your position in this organization at the earliest. Hope this helps!