parametrize test, and a note to v3.0.0

Author: natmokval

doc/source/whatsnew/v3.0.0.rst

@@ -456,6 +456,7 @@ Datetimelike
 - Bug in :func:`tseries.api.guess_datetime_format` would fail to infer time format when "%Y" == "%H%M" (:issue:`57452`)
 - Bug in :meth:`DatetimeIndex.is_year_start` and :meth:`DatetimeIndex.is_quarter_start` does not raise on Custom business days frequencies bigger then "1C" (:issue:`58664`)
 - Bug in :meth:`DatetimeIndex.is_year_start` and :meth:`DatetimeIndex.is_quarter_start` returning ``False`` on double-digit frequencies (:issue:`58523`)
+- Bug in :meth:`DatetimeIndex.is_year_start` and :meth:`DatetimeIndex.is_quarter_start` where using Business month begin frequency (:issue:`58729`)
 - Bug in setting scalar values with mismatched resolution into arrays with non-nanosecond ``datetime64``, ``timedelta64`` or :class:`DatetimeTZDtype` incorrectly truncating those scalars (:issue:`56410`)
 
 Timedelta

pandas/tests/indexes/datetimes/test_scalar_compat.py

@@ -344,6 +344,11 @@ def test_dti_is_year_start_freq_custom_business_day_with_digit(self):
         with pytest.raises(ValueError, match=msg):
             dr.is_year_start
 
-    def test_dti_is_year_start_freq_business_month_start_with(self):
-        dr = date_range("2020-01-01", periods=1, freq="BMS")
+    @pytest.mark.parametrize("freq", ["BMS", offsets.BusinessMonthBegin()])
+    def test_dti_is_year_quarter_start_freq_business_month_begin(self, freq):
+        # GH#58729
+        dr = date_range("2020-01-01", periods=1, freq=freq)
         assert all(dr.is_year_start)
+
+        dr = date_range("2020-01-01", periods=1, freq=freq)
+        assert all(dr.is_quarter_start)