statistics in zonerunner

What you really want to enable is the feature of DNS query/response logging. However, you would want to use High Speed Logging (HSL) and an external syslog type of solution like Splunk or LogRhytm. Reference: https://my.f5.com/manage/s/article/K65762138

I tried to see if there were any TMSH commands or SNMP OIDs that could be use to obtain this info, but no dice.

HOWEVER! There is the AVR module to the rescue ;) Please see the following: https://techdocs.f5.com/kb/en-us/products/big-ip-dns/manuals/product/bigip-dns-services-implementations-12-1-0/12.html. You can then do fancy commands like the following:

show analytics dns report view-by query-name drilldown { { entity query-type values {A}}} limit 3

I would give that a shot as a quick win. Make sure you have enough system resources if you need to enable AVR and remember that when enabling, services will restart. So make sure you have change control if doing this on a production system.