diff --git a/doc/source/whatsnew/v2.2.0.rst b/doc/source/whatsnew/v2.2.0.rst
index 2bc031e4b5f61..38f39270d12c7 100644
--- a/doc/source/whatsnew/v2.2.0.rst
+++ b/doc/source/whatsnew/v2.2.0.rst
@@ -103,6 +103,7 @@ Other enhancements
 - :meth:`ExtensionArray.duplicated` added to allow extension type implementations of the ``duplicated`` method (:issue:`55255`)
 - DataFrame.apply now allows the usage of numba (via ``engine="numba"``) to JIT compile the passed function, allowing for potential speedups (:issue:`54666`)
 - Implement masked algorithms for :meth:`Series.value_counts` (:issue:`54984`)
+- Improved error message when constructing :class:`Period` with invalid offsets such as "QS" (:issue:`55785`)
 
 .. ---------------------------------------------------------------------------
 .. _whatsnew_220.notable_bug_fixes:
diff --git a/pandas/_libs/tslibs/offsets.pyx b/pandas/_libs/tslibs/offsets.pyx
index df3a2e3ecde48..0150aeadbd0ab 100644
--- a/pandas/_libs/tslibs/offsets.pyx
+++ b/pandas/_libs/tslibs/offsets.pyx
@@ -4177,7 +4177,7 @@ cdef class CustomBusinessDay(BusinessDay):
     def _period_dtype_code(self):
         # GH#52534
         raise TypeError(
-            "CustomBusinessDay cannot be used with Period or PeriodDtype"
+            "CustomBusinessDay is not supported as period frequency"
         )
 
     _apply_array = BaseOffset._apply_array
diff --git a/pandas/_libs/tslibs/period.pyx b/pandas/_libs/tslibs/period.pyx
index 700f500c1d8b8..d1f925f3a0b48 100644
--- a/pandas/_libs/tslibs/period.pyx
+++ b/pandas/_libs/tslibs/period.pyx
@@ -2726,6 +2726,14 @@ class Period(_Period):
 
         if freq is not None:
             freq = cls._maybe_convert_freq(freq)
+            try:
+                period_dtype_code = freq._period_dtype_code
+            except (AttributeError, TypeError):
+                # AttributeError: _period_dtype_code might not exist
+                # TypeError: _period_dtype_code might intentionally raise
+                raise TypeError(
+                    f"{(type(freq).__name__)} is not supported as period frequency"
+                )
         nanosecond = 0
 
         if ordinal is not None and value is not None:
@@ -2758,7 +2766,7 @@ class Period(_Period):
 
         elif is_period_object(value):
             other = value
-            if freq is None or freq._period_dtype_code == other._dtype._dtype_code:
+            if freq is None or period_dtype_code == other._dtype._dtype_code:
                 ordinal = other.ordinal
                 freq = other.freq
             else:
diff --git a/pandas/tests/dtypes/test_dtypes.py b/pandas/tests/dtypes/test_dtypes.py
index 27994708d2bdb..04b3feabb9854 100644
--- a/pandas/tests/dtypes/test_dtypes.py
+++ b/pandas/tests/dtypes/test_dtypes.py
@@ -444,7 +444,7 @@ def test_construction(self):
 
     def test_cannot_use_custom_businessday(self):
         # GH#52534
-        msg = "CustomBusinessDay cannot be used with Period or PeriodDtype"
+        msg = "CustomBusinessDay is not supported as period frequency"
         msg2 = r"PeriodDtype\[B\] is deprecated"
         with pytest.raises(TypeError, match=msg):
             with tm.assert_produces_warning(FutureWarning, match=msg2):
diff --git a/pandas/tests/indexes/period/test_constructors.py b/pandas/tests/indexes/period/test_constructors.py
index f1db5ab28be30..505050a2089d8 100644
--- a/pandas/tests/indexes/period/test_constructors.py
+++ b/pandas/tests/indexes/period/test_constructors.py
@@ -553,6 +553,21 @@ def test_map_with_string_constructor(self):
         # lastly, values should compare equal
         tm.assert_index_equal(res, expected)
 
+    @pytest.mark.parametrize(
+        "freq, freq_msg",
+        [
+            (offsets.BYearBegin(), "BYearBegin"),
+            (offsets.YearBegin(2), "YearBegin"),
+            (offsets.QuarterBegin(startingMonth=12), "QuarterBegin"),
+            (offsets.BusinessMonthEnd(2), "BusinessMonthEnd"),
+        ],
+    )
+    def test_offsets_not_supported(self, freq, freq_msg):
+        # GH#55785
+        msg = f"{freq_msg} is not supported as period frequency"
+        with pytest.raises(TypeError, match=msg):
+            Period(year=2014, freq=freq)
+
 
 class TestShallowCopy:
     def test_shallow_copy_empty(self):
diff --git a/pandas/tests/scalar/period/test_asfreq.py b/pandas/tests/scalar/period/test_asfreq.py
index 597282e10052e..7164de0a228d9 100644
--- a/pandas/tests/scalar/period/test_asfreq.py
+++ b/pandas/tests/scalar/period/test_asfreq.py
@@ -825,7 +825,8 @@ def test_asfreq_MS(self):
         with pytest.raises(ValueError, match=msg):
             initial.asfreq(freq="MS", how="S")
 
-        with pytest.raises(ValueError, match=msg):
+        msg = "MonthBegin is not supported as period frequency"
+        with pytest.raises(TypeError, match=msg):
             Period("2013-01", "MS")
 
         assert _period_code_map.get("MS") is None
diff --git a/pandas/tests/scalar/period/test_period.py b/pandas/tests/scalar/period/test_period.py
index 8cc3ace52a4d4..448c2091e14f6 100644
--- a/pandas/tests/scalar/period/test_period.py
+++ b/pandas/tests/scalar/period/test_period.py
@@ -38,7 +38,7 @@
 class TestPeriodConstruction:
     def test_custom_business_day_freq_raises(self):
         # GH#52534
-        msg = "CustomBusinessDay cannot be used with Period or PeriodDtype"
+        msg = "CustomBusinessDay is not supported as period frequency"
         with pytest.raises(TypeError, match=msg):
             Period("2023-04-10", freq="C")
         with pytest.raises(TypeError, match=msg):
@@ -1628,8 +1628,8 @@ def test_negone_ordinals():
 
 
 def test_invalid_frequency_error_message():
-    msg = "Invalid frequency: <WeekOfMonth: week=0, weekday=0>"
-    with pytest.raises(ValueError, match=msg):
+    msg = "WeekOfMonth is not supported as period frequency"
+    with pytest.raises(TypeError, match=msg):
         Period("2012-01-02", freq="WOM-1MON")