It's not possible to change the implementation of the internal timestamps recorded by dbt's artifacts (i.e. the timestamps recorded in the on-run-end context results var or the run_results.json file) - so you'll have to manually change it yourself using some builtin modules. Keep in mind that this does not change the original values in run_results.json or in results - those WILL always be in UTC timestamps.
In the example below, we're simply converting one of the timestamps dbt records (when a model is completed) into a different timezone - then it is our responsiblity for then taking that converted value and writing it to a table in the database if we want to.
# dbt_project.yml
name: my_dbt_project
profile: all
config-version: 2
version: 1.0
models:
my_dbt_project:
+materialized: table
on-run-end: "{{ check_results(results) }}"-- models/foo.sql
select 1 id
-- macros/check_results.sql
{% macro check_results(results) -%}
{% if execute %}
{% for r in results %}
{% set local_tz = modules.pytz.timezone("Pacific/Auckland") %}
{% set ts_raw = r.timing[1].completed_at %}
{% set ts_utc_aware = modules.pytz.utc.localize(ts_raw) %}
{% set ts_in_local_tz = ts_utc_aware.astimezone(local_tz) %}
{% set line %}
-----------------------------------------------------------
node {{ r.node.unique_id }}
status: {{ r.status }}
completed_at_utc: {{ ts_utc_aware }}
completed_at_local: {{ ts_in_local_tz }}
-----------------------------------------------------------
{% endset %}
{% do log(line, True) %}
{% endfor %}
{% endif %}
{%- endmacro %}$ dbt run
05:01:44 Running with dbt=1.6.3
05:01:45 Registered adapter: snowflake=1.6.2
05:01:46 Found 1 model, 1 operation, 0 sources, 0 exposures, 0 metrics, 373 macros, 0 groups, 0 semantic models
05:01:46
05:01:52 Concurrency: 1 threads (target='sf')
05:01:52
05:01:52 1 of 1 START sql table model dbt_jyeo.foo ...................................... [RUN]
05:01:55 1 of 1 OK created sql table model dbt_jyeo.foo ................................. [SUCCESS 1 in 3.85s]
05:01:55
05:01:55 Running 1 on-run-end hook
05:01:55
-----------------------------------------------------------
node model.my_dbt_project.foo
status: success
completed_at_utc: 2023-09-22 05:01:55.337792+00:00
completed_at_local: 2023-09-22 17:01:55.337792+12:00
-----------------------------------------------------------
05:01:55 1 of 1 START hook: my_dbt_project.on-run-end.0 ................................. [RUN]
05:01:55 1 of 1 OK hook: my_dbt_project.on-run-end.0 .................................... [OK in 0.00s]
05:01:55
05:01:55
05:01:55 Finished running 1 table model, 1 hook in 0 hours 0 minutes and 9.45 seconds (9.45s).
05:01:55
05:01:55 Completed successfully
05:01:55
05:01:55 Done. PASS=1 WARN=0 ERROR=0 SKIP=0 TOTAL=1