diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index 1ddefc94bcbad..12a6fc4d52053 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -914,6 +914,7 @@ Datetimelike
 - Bug in constructing a :class:`DataFrame` or :class:`Series` with mismatched ``datetime64`` data and ``timedelta64`` dtype, or vice-versa, failing to raise a ``TypeError`` (:issue:`38575`, :issue:`38764`, :issue:`38792`)
 - Bug in constructing a :class:`Series` or :class:`DataFrame` with a ``datetime`` object out of bounds for ``datetime64[ns]`` dtype or a ``timedelta`` object out of bounds for ``timedelta64[ns]`` dtype (:issue:`38792`, :issue:`38965`)
 - Bug in :meth:`DatetimeIndex.intersection`, :meth:`DatetimeIndex.symmetric_difference`, :meth:`PeriodIndex.intersection`, :meth:`PeriodIndex.symmetric_difference` always returning object-dtype when operating with :class:`CategoricalIndex` (:issue:`38741`)
+- Bug in :meth:`DatetimeIndex.intersection` giving incorrect results with non-Tick frequencies with ``n != 1`` (:issue:`42104`)
 - Bug in :meth:`Series.where` incorrectly casting ``datetime64`` values to ``int64`` (:issue:`37682`)
 - Bug in :class:`Categorical` incorrectly typecasting ``datetime`` object to ``Timestamp`` (:issue:`38878`)
 - Bug in comparisons between :class:`Timestamp` object and ``datetime64`` objects just outside the implementation bounds for nanosecond ``datetime64`` (:issue:`39221`)
diff --git a/pandas/core/indexes/datetimelike.py b/pandas/core/indexes/datetimelike.py
index 53ab2e45edede..7143b275214ee 100644
--- a/pandas/core/indexes/datetimelike.py
+++ b/pandas/core/indexes/datetimelike.py
@@ -693,7 +693,8 @@ def _can_fast_intersect(self: _T, other: _T) -> bool:
         elif self.freq.is_anchored():
             # this along with matching freqs ensure that we "line up",
             #  so intersection will preserve freq
-            return True
+            # GH#42104
+            return self.freq.n == 1
 
         elif isinstance(self.freq, Tick):
             # We "line up" if and only if the difference between two of our points
@@ -702,7 +703,8 @@ def _can_fast_intersect(self: _T, other: _T) -> bool:
             remainder = diff % self.freq.delta
             return remainder == Timedelta(0)
 
-        return True
+        # GH#42104
+        return self.freq.n == 1
 
     def _can_fast_union(self: _T, other: _T) -> bool:
         # Assumes that type(self) == type(other), as per the annotation
diff --git a/pandas/tests/indexes/datetimes/test_setops.py b/pandas/tests/indexes/datetimes/test_setops.py
index 513a47d6be7ab..62663c8c6b810 100644
--- a/pandas/tests/indexes/datetimes/test_setops.py
+++ b/pandas/tests/indexes/datetimes/test_setops.py
@@ -391,6 +391,23 @@ def test_setops_preserve_freq(self, tz):
         assert result.freq == rng.freq
         assert result.tz == rng.tz
 
+    def test_intersection_non_tick_no_fastpath(self):
+        # GH#42104
+        dti = DatetimeIndex(
+            [
+                "2018-12-31",
+                "2019-03-31",
+                "2019-06-30",
+                "2019-09-30",
+                "2019-12-31",
+                "2020-03-31",
+            ],
+            freq="Q-DEC",
+        )
+        result = dti[::2].intersection(dti[1::2])
+        expected = dti[:0]
+        tm.assert_index_equal(result, expected)
+
 
 class TestBusinessDatetimeIndex:
     def setup_method(self, method):