Our clients will send two numbers to this server-side compute engine, and we want this server to notify all those clients who sent numbers that add up > 10.
Researching online it seems Google Compute Engine can do this but we’re not sure if it can send notifications.
This is our first time setting up something like this so we are confused how to go about it
As far as I understood you want a simple push notification mechanism. Firebase is part of the GCP so when you decide to use Firebase, actually you are using a GCP project.
Consider taking a look at this link to understand their relationship better.
On the other hand, there is no more separate cloud messaging only for GCP according to this. Now your only option if you want to stay faithful to google platform for that matter is Firebase Cloud Messaging