Constant writing operations on google cloud database
For a university course in Data Visualisation, I used the Prometheus/Grafana stack  (great s/ware) to present images of various database metrics while running various HammerDB benchmarks (also an excellent tool).
As part of this, to provide a baseline, I looked at my (single) server &quot;at rest&quot; and noticed (as you have) that there was activity - albeit a very small amount (~ 10 writes/second) - on an &quot;inactive&quot; server.
Basically, this is &quot;house-keeping&quot; functionality which happens even if there is no ongoing external DDL or DML.
There is an interesting article here and here (more recent and probably more accurate) showing the impact of the number of idle connections on the NOTPM (No. of Transactions Per Minute) results from the DBT2 benchmark tool.
As you can see from the results displayed in the second article, there is a small (but not non-existent) effect from just having idle connections.
You say that you are running a cluster - I would imagine that there is a  communications overhead there with nodes pinging each other even in the absence of any serious workload (or even any workload at all).
You don't mention what cluster (NDB? Galera?), but I'm sure that a delve into the code (way above my pay grade... :-) ), a search, or a question on your cluster's architecture/guru list would be able to provide more detail.
If you're interested and have the time and the inclination to test what's going on, you may like to leave your &quot;inactive&quot; cluster running for a while (say over a weekend?) and run (some variant of) this query before and after you begin the test (adapted from here):
SELECT 
     table_schema AS the_db, 
     table_name AS the_table, 
     round(((data_length + index_length) / 1024 / 1024), 2) `size_in_mb` 
FROM information_schema.TABLES 
ORDER BY (data_length + index_length) DESC;

You might like to include the information_schema (and the mysql schema) as well since some/much/all of this activity that you are seeing might be occurring there.
This would presumably be dependent on the architecture of your cluster. Inserting the results of these queries into tables of their own should render any comparisons which you may wish to do relatively trivial.
Also, bear in mind that there may not be a whole lot to see, as these writes may simply be UPDATEs of a single or just a few records?
p.s. welcome to the forum and +1 for an interesting question!