Quorum won't solve consistency problems like this unless you can also guarantee a serialized reader/writer for any given piece of data. The best you can do by pinning queries to primary replicas and enforcing R + W > N is RYOW consistency, and that depends on the aforementioned serialization point.
Quorum reads + writes and applying non-destructive updates only are enough to get linearizabile consistency in Cassandra, even if clocks are not perfectly synchronized.
If you're applying non-destructive updates without quorum, you would not get monotonic read consistency, because stale reads would be possible. You could write a row, immediately read it back from another replica and find out the row is not there, because that replica didn't get it yet. You'd have only eventual consistency.
